I've owned the Sigma 150-600 F5.0-6.3 Contemporary for a while now and just yesterday bought the Canon 100-400 4.5-5.6L II. The main purpose of both lenses is birding and backyard wildlife.
I thought I'd share my thoughts comparing the two. Originally, I bought the Sigma 150-600C first because of price and reach, but really, I agonized over it. Hopefully, this may help someone else in their decision-making.
First of all, since it was way too late to do anything with my shiny new Canon 100-400 II last night by the time I got home, I set up AFMA, and then decided to test it for image quality versus the Sigma, in my basement. I set up a flattened calibration target on one side, and an Xrite Colorchecker on the other, like so:
To make it apples to apples, they're both photographed at 400mm f/8. Ultimately, this helps the Sigma more than it helps the Canon, as the Sigma's IQ suffers a little wide open. There are a zillion reviews where you can see by how much; I think it's relatively inconsequential, though.
Here are the full resolution version versions, unedited. These had CWB set against the ColorChecker White Card, but they are otherwise uncorrected. They were originally shot as RAW, and converted in Photoshop using ACR with no manipulation.
Canon: http://talys.icxi.com/canon100-400/IMG_7995_Canon_100-400II.jpg
Sigma: http://talys.icxi.com/canon100-400/IMG_8008_Sigma_150-600C.jpg
For sharpness, I think that they are very, very close. I wouldn't say that one is sharper than the other. However, if you look at the ruled target on the left with the two side by side:
The Canon has visibly better contrast, with whiter whites. We can haggle about the definition of "sharpness" or "pixel level sharpnes" as either bleed or edge contrast, but at the end of the day, cleaner whites and more contrast will appear more sharp in a general sense. And besides, a whiter white is a better thing, so I give the Canon 100-400L II the win over the Sigma 150-600C.
Note that on the target, the print is not a solid black, but a dark grey dithered pattern, which is accurately represented on both lenses.
Interesting aside: When I was setting up and checking focus and exposure, I had my camera set to JPEG (for faster WiFi, since I was verifying focus on PC). The Sigma exhibited a slight, but very unappealing moiré. It didn't appear in any of the RAWs. Just for kicks, I tried it on the Canon, and there was no moiré. However, I always shoot RAW, so this doesn't really matter to me.
Next, here's the ColorCheckr side by side. However, you should use this image only for judging the colors and contrast, as they have both been rotated to be square and then resized. To judge the sharpness, use the original images above, which aren't rotated.
In unedited photos, I favor the Canon, which looks a little more contrasty to me, with a nicer red, green, and blue, but that's probably just taste. When you apply the Xrite corrections, they become indistinguishable, and in the real world, whatever you do in post is probably going to have infinitely more impact.
I will give you some of my thoughts between using both, but before I get to that, I'll share my first pictures -- some animal pictures I shot today, since you made it this far through my rambling
All of these were shot with the Canon 100-400 4.5-5.6L II, on a Canon 6D Mark II. The high res images are cropped, but unreduced, so that you can pixel peep if you wish. I was pretty impressed with the ISO 12,800 shot at the end.
Raccoon Family Photo, 1/800 f/5 ISO 4000 135mm, High Res: http://talys.icxi.com/canon100-400/Family_Photo_F.JPG
Raccoon Baby, 1/400 f/5 ISO 2500 300mm, High Res: http://talys.icxi.com/canon100-400/Baby3_F.jpg
Chickadee, 1/2000 f/5.6 ISO 3200 214mm, High Res: http://talys.icxi.com/canon100-400/Chickadee_F.jpg
Chickadee Launch, 1/2000 f/5.6 ISO 3200 200mm
Chickadee Incoming, 1/1250, f/5.6 ISO 12800 400mm, High Res: http://talys.icxi.com/canon100-400/Chickadee_Launch_F.jpg
Now, having used both, here are some thoughts.
Things I liked better about the Canon 100-400L II:
- The Canon 100-400L II is a joy to use. The construction is amazing, definitely a step up from the Sigma 150-600 C in the department of premium feel, first impression, and construction. It's the difference between very nice and awesome, though, so it's not like I don't like the Sigma. I just like the Canon more.
- The weight is amazing. It's a well-balanced lens that is great for handheld shots. The Sigma 150-600 is a hard lens to use handheld for a half-day, because it gets very heavy -- around twice the weight of the Canon. This, and focal range, are obviously the two biggest differences between the two lenses.
- The minimum focus distance of the Canon is out of this world short. 3.5 ft versus about 6.5 ft on the Sigma -- approximately, anyhow; I'm going by memory. But I REALLY noticed this, and the use cases that it opens up is pretty huge.
- 100-150 is a very useful focal range.
- The zoom lock lets you lock the lens in place at any zoom level. It's super useful, because you can extend the zoom to 250, 300, whatever -- and keep it there without it creeping to full extension, especially if you walk around with a harness.
- The manual focus ring is superb. On a scale of 10, the Canon is a perfect 10, and the Sigma MF is a super annoying 5, and that's proably being generous. The Sigma ring is very, very narrow (only a few millimeters), relatively stiff, and quite jumpy (tiniest twist can throw focus off). On the other hand, the Canon is butter smooth, responsively fast without being jerky, and very comfortable to use.
- Speaking of focus, Autofocus is very good on the Canon, definitely far superior to the Sigma. At 6.3, the Sigma still has trouble at times; you really need to be wide open and zoomed out to get focus, and even then, the AF speed isn't comparable.
- IS Mode 3 is available, which makes it so that IS only kicks in the moment the shutter is going to open. Cool feature, normally on only Big White Primes. It is useful in that you don't have to fight IS to focus.
- Both the Canon and Sigma come with very nice lens hoods. Yeah, it's just a lens hood, but still. The Canon one has a little door at the bottom, which is really sweet -- you can adjust polarizing/ND filters without taking the hood off.
Things I liked better about the Sigma 150-600 Contemporary:
- The elephant in the room is the price, of course. It's one third the price of the Canon. The Sigma is much better priced for casual use. (pricing note: for some reason, the Sigma is cheaper in Canada. It comes out to about $800 USD, versus about $1,000 USD in the US; the Canon, on the other hand is a about $2,050 USD at the current manufacturer's rebate, and I believe $150 or so without).
- And the other elephant in the room is the focal range. 150-600 is massively more. Now, to be fair, the Canon can mitigate this by adding a 1.4x TC, taking it to 140mm-560mm. Thing is, the 1.4TC is about half the price of the Sigma.
- After using both, I prefer the tripod collar on the Sigma. It can be easily and quickly removed and is easier to level because of intelligent markings on the collar and lens.
- You can update the firmware on the Sigma with the USB dock and you can configure several AF parameters in C1, C2. Plus, you can adjust AF from the USB dock, if you're using a body with AFMA.
CONCLUSION
I was really impressed by how well the Sigma held up!
Personally, I see these as complimentary lenses: the Sigma is really too heavy for me to do handheld photography (at least for very long), so it's generally always on a tripod. That gives me the ability to shoot lower ISOs and a little slower shutter speed, which goes well with being on 80D/APSC -- which also gives me more reach (nearly 1000mm of FF equivalent magnification).
For the Canon 100-400 II, I plan on attaching it to my 6DII to catch birds in flight. Here, high shutter speeds are really important, and because of, well, math, there aren't many ways to shoot 1/2000 at an APSC usable ISO.
On my outings, I plan on taking both, wearing them on a BlackRapid dual harness.
So if I had it to do again, which one would I buy first? Even though I think I like the Canon lens better, I think I would still buy the Sigma first. It is an amazing value, and nearly every shot you can get from a Canon 100-400L II, you could get on a Sigma... it's just that some would be a lot harder.
Which would buy if I could only buy one? Well, in this case, I'd get the Canon, or if I already had the Sigma, I'd sell that and get the Canon (plus TC if I needed the reach), when finances permitted. Ultimately, it's just too good a lens to not own, if you don't think that $2000 is too much to spend on an up-to-400mm telephoto.
But at the end of the day, it depends on what you want to shoot. I think if most of your shooting is handheld, you should get the Canon 100-400 4.5-5.6L II, if you can afford it. If not, I'd seriously suggest looking into the new Sigma 100-400, because it's really very impressive, and also a fraction of the Canon's price. If you're going to shoot everything on a tripod, I think that the Sigma 150-600 5.0-6.3 Contemporary is the winner.
Either way, they are both great lenses, and I'm very happy to own both!
I thought I'd share my thoughts comparing the two. Originally, I bought the Sigma 150-600C first because of price and reach, but really, I agonized over it. Hopefully, this may help someone else in their decision-making.
First of all, since it was way too late to do anything with my shiny new Canon 100-400 II last night by the time I got home, I set up AFMA, and then decided to test it for image quality versus the Sigma, in my basement. I set up a flattened calibration target on one side, and an Xrite Colorchecker on the other, like so:

To make it apples to apples, they're both photographed at 400mm f/8. Ultimately, this helps the Sigma more than it helps the Canon, as the Sigma's IQ suffers a little wide open. There are a zillion reviews where you can see by how much; I think it's relatively inconsequential, though.
Here are the full resolution version versions, unedited. These had CWB set against the ColorChecker White Card, but they are otherwise uncorrected. They were originally shot as RAW, and converted in Photoshop using ACR with no manipulation.
Canon: http://talys.icxi.com/canon100-400/IMG_7995_Canon_100-400II.jpg
Sigma: http://talys.icxi.com/canon100-400/IMG_8008_Sigma_150-600C.jpg
For sharpness, I think that they are very, very close. I wouldn't say that one is sharper than the other. However, if you look at the ruled target on the left with the two side by side:

The Canon has visibly better contrast, with whiter whites. We can haggle about the definition of "sharpness" or "pixel level sharpnes" as either bleed or edge contrast, but at the end of the day, cleaner whites and more contrast will appear more sharp in a general sense. And besides, a whiter white is a better thing, so I give the Canon 100-400L II the win over the Sigma 150-600C.
Note that on the target, the print is not a solid black, but a dark grey dithered pattern, which is accurately represented on both lenses.
Interesting aside: When I was setting up and checking focus and exposure, I had my camera set to JPEG (for faster WiFi, since I was verifying focus on PC). The Sigma exhibited a slight, but very unappealing moiré. It didn't appear in any of the RAWs. Just for kicks, I tried it on the Canon, and there was no moiré. However, I always shoot RAW, so this doesn't really matter to me.
Next, here's the ColorCheckr side by side. However, you should use this image only for judging the colors and contrast, as they have both been rotated to be square and then resized. To judge the sharpness, use the original images above, which aren't rotated.

In unedited photos, I favor the Canon, which looks a little more contrasty to me, with a nicer red, green, and blue, but that's probably just taste. When you apply the Xrite corrections, they become indistinguishable, and in the real world, whatever you do in post is probably going to have infinitely more impact.
I will give you some of my thoughts between using both, but before I get to that, I'll share my first pictures -- some animal pictures I shot today, since you made it this far through my rambling
All of these were shot with the Canon 100-400 4.5-5.6L II, on a Canon 6D Mark II. The high res images are cropped, but unreduced, so that you can pixel peep if you wish. I was pretty impressed with the ISO 12,800 shot at the end.
Raccoon Family Photo, 1/800 f/5 ISO 4000 135mm, High Res: http://talys.icxi.com/canon100-400/Family_Photo_F.JPG

Raccoon Baby, 1/400 f/5 ISO 2500 300mm, High Res: http://talys.icxi.com/canon100-400/Baby3_F.jpg

Chickadee, 1/2000 f/5.6 ISO 3200 214mm, High Res: http://talys.icxi.com/canon100-400/Chickadee_F.jpg

Chickadee Launch, 1/2000 f/5.6 ISO 3200 200mm

Chickadee Incoming, 1/1250, f/5.6 ISO 12800 400mm, High Res: http://talys.icxi.com/canon100-400/Chickadee_Launch_F.jpg
Now, having used both, here are some thoughts.
Things I liked better about the Canon 100-400L II:
- The Canon 100-400L II is a joy to use. The construction is amazing, definitely a step up from the Sigma 150-600 C in the department of premium feel, first impression, and construction. It's the difference between very nice and awesome, though, so it's not like I don't like the Sigma. I just like the Canon more.
- The weight is amazing. It's a well-balanced lens that is great for handheld shots. The Sigma 150-600 is a hard lens to use handheld for a half-day, because it gets very heavy -- around twice the weight of the Canon. This, and focal range, are obviously the two biggest differences between the two lenses.
- The minimum focus distance of the Canon is out of this world short. 3.5 ft versus about 6.5 ft on the Sigma -- approximately, anyhow; I'm going by memory. But I REALLY noticed this, and the use cases that it opens up is pretty huge.
- 100-150 is a very useful focal range.
- The zoom lock lets you lock the lens in place at any zoom level. It's super useful, because you can extend the zoom to 250, 300, whatever -- and keep it there without it creeping to full extension, especially if you walk around with a harness.
- The manual focus ring is superb. On a scale of 10, the Canon is a perfect 10, and the Sigma MF is a super annoying 5, and that's proably being generous. The Sigma ring is very, very narrow (only a few millimeters), relatively stiff, and quite jumpy (tiniest twist can throw focus off). On the other hand, the Canon is butter smooth, responsively fast without being jerky, and very comfortable to use.
- Speaking of focus, Autofocus is very good on the Canon, definitely far superior to the Sigma. At 6.3, the Sigma still has trouble at times; you really need to be wide open and zoomed out to get focus, and even then, the AF speed isn't comparable.
- IS Mode 3 is available, which makes it so that IS only kicks in the moment the shutter is going to open. Cool feature, normally on only Big White Primes. It is useful in that you don't have to fight IS to focus.
- Both the Canon and Sigma come with very nice lens hoods. Yeah, it's just a lens hood, but still. The Canon one has a little door at the bottom, which is really sweet -- you can adjust polarizing/ND filters without taking the hood off.
Things I liked better about the Sigma 150-600 Contemporary:
- The elephant in the room is the price, of course. It's one third the price of the Canon. The Sigma is much better priced for casual use. (pricing note: for some reason, the Sigma is cheaper in Canada. It comes out to about $800 USD, versus about $1,000 USD in the US; the Canon, on the other hand is a about $2,050 USD at the current manufacturer's rebate, and I believe $150 or so without).
- And the other elephant in the room is the focal range. 150-600 is massively more. Now, to be fair, the Canon can mitigate this by adding a 1.4x TC, taking it to 140mm-560mm. Thing is, the 1.4TC is about half the price of the Sigma.
- After using both, I prefer the tripod collar on the Sigma. It can be easily and quickly removed and is easier to level because of intelligent markings on the collar and lens.
- You can update the firmware on the Sigma with the USB dock and you can configure several AF parameters in C1, C2. Plus, you can adjust AF from the USB dock, if you're using a body with AFMA.
CONCLUSION
I was really impressed by how well the Sigma held up!
Personally, I see these as complimentary lenses: the Sigma is really too heavy for me to do handheld photography (at least for very long), so it's generally always on a tripod. That gives me the ability to shoot lower ISOs and a little slower shutter speed, which goes well with being on 80D/APSC -- which also gives me more reach (nearly 1000mm of FF equivalent magnification).
For the Canon 100-400 II, I plan on attaching it to my 6DII to catch birds in flight. Here, high shutter speeds are really important, and because of, well, math, there aren't many ways to shoot 1/2000 at an APSC usable ISO.
On my outings, I plan on taking both, wearing them on a BlackRapid dual harness.
So if I had it to do again, which one would I buy first? Even though I think I like the Canon lens better, I think I would still buy the Sigma first. It is an amazing value, and nearly every shot you can get from a Canon 100-400L II, you could get on a Sigma... it's just that some would be a lot harder.
Which would buy if I could only buy one? Well, in this case, I'd get the Canon, or if I already had the Sigma, I'd sell that and get the Canon (plus TC if I needed the reach), when finances permitted. Ultimately, it's just too good a lens to not own, if you don't think that $2000 is too much to spend on an up-to-400mm telephoto.
But at the end of the day, it depends on what you want to shoot. I think if most of your shooting is handheld, you should get the Canon 100-400 4.5-5.6L II, if you can afford it. If not, I'd seriously suggest looking into the new Sigma 100-400, because it's really very impressive, and also a fraction of the Canon's price. If you're going to shoot everything on a tripod, I think that the Sigma 150-600 5.0-6.3 Contemporary is the winner.
Either way, they are both great lenses, and I'm very happy to own both!