I had the same question. I own the old 70-300 IS and it really lacks IQ at the long end. After watching a youtube of FroKnowsPhoto review of the 70-300L he raises some serious questions.
In the video he is snapping shots of a kid's soccer game. While the pics are tac sharp and the lens is very good in quality, the slow aperture at longer distances limits the amount you can "blow out" the backgrounds.
Both lens will have similar IQ in my opinion. A little different purpose though. I am going with the 70-200 F/4.0L because: 1) seems better for portraits (faster aperture for DOF effects) and 2) ability to use TC.
I have the 70-200mm f/2.8 IS L II and the 70-300mm f/4-5.6 IS L, along with the 1.4x tele-extender III.
Couple of comments on the above:
The tele-extender is going to slow down your AF by 50% and you will notice this on the OP's 550D. While faster cameras like the 5D3 and the 1Dx can compensate due to their AF systems speed, I can say the combination of the 70-200mm f/2.8 IS rev2 and the 1.4x tele-extender rev3 on the 7D will at times drift off an AF lock when shooting sports like equestrian jumping or professional cycling. It's not bad, but it does happen, at 8 fps I would say 20% of images can be soft on the focus with this combo. However, when it's locked, the images are far sharper than the older version combination of that lens and TC.
On the 5D3 the slowness isn't really much of an impact due to the speed of the AF system in the first place, but I've had the 5D3 for a month not and have not shot that combo much. Am waiting on a 1Dx, so I can't comment on that combo.
On the "blow out the background"
comment regarding the 70-300mm f/4-5.6 IS L, at full zoom focal length, while you're not going to get the kind of bokeh you would see at f/4 or at f/2.8 with the pricier 70-200mm, the DOF at 300mm, f/5.6 with a target 25 yards away (75 ft, or about 23 meters) on a crop body is 4.1 feet (about 1.3 meters).
To put that in perspective, that is shooting at an effective 480mm and reaching out to the middle of a football field. That 4.1ft DOF isn't as much as you think it is and you can blow out or blur the background very acceptably on the opposite sidelines.
Performance wise, the 70-300mm does well, it's obviously lighter than the 70-200mm f/2.8 and thus easier to manage, the IS performs very well, it has a LOCK switch to stop the lens from being able to zoom or creep if the weight causes it to do so when hanging down. It's fast, the AF is zippy on the 7D and the 5D2 and 5D3.Negatives on the 70-300mm:
1) slower aperture - yes, if you need the extra stops, this won't be the right lens. However, I've shot this at weddings and receptions with and without flash at ISO's up to 3200 on a 7D without issue. The reach on a 7D in a church or outdoor wedding outweighs the slower aperture when you need focal length.
2) The zoom and manual focus rings are reversed on this lens when compared to the 70-200, the 24-70, the 24-105, 16-35, etc..... In other words, the far ring is the zoom, the near ring is the manual focus. This is frustrating, for one, it breaks what you've learned to do with the zoom ring as you find yourself reaching for the wrong ring at times. Second, it makes you extend your arm further from your body, leading to a more unstable stance when shooting. I like to keep my arm close and tight to solidify my stance. On the 70-300 you have to move your arm further out to handle the zoom ring. It's a PITA, I have no idea why Canon did that. It's the #1 reason I could dump the lens if a new 100-400mm f/4-5.6 IS L were to appear.
3) The hood, filter mount and lens extension....the hood "looks" cheap....no fancy tulip design, it's a straight up barrel. It just looks dumpy...but it works. The filter mount is 67mm which means it does not fit the filters I use with the 70-200mm or several other lenses I have...so if you use polarizers or ND's, just know you might be buying more to fit it. Note the lens is not internal focus and extends 2.11" at 300mm full zoom.
4) Variable aperture - you need to account for that in your exposures, especially when shooting in manual mode and tracking/zooming in-out at the same time. The lens max apertures by natural focal length (unadjusted for crop 1.6x body) are:
Overall, if you can accept the slower AF of the tele-extender combo and more chance of blowing the focus lock, along with the additional but very slight softness it adds (assuming you have the rev3 extender which is way better than the rev2) I would go with the70-200 f/4 and extender....you have greater flexibility.
If you want a solid, light weight, and sharp solution, the 70-300mm is a great choice, assuming you can deal with the reversed focus/zoom rings (god I hate that!!!), this is a great choice.
Frankly, I bought it to use on the 5D3 and the 1Dx in order to recover some of the focal length lost on a full frame body, when compared with the reach the 7D had with the 70-200mm. I'm selling my 5D2 and my 7D once I have the 1Dx in and I've become comfortable with the 5D3. As such, I wanted to recover some reach I'll lose when the 7D is gone, and I didn't want to have to rely on the 1.4x extender on the tele-zooms all the time. I didn't want the current 100-400mm and with no new version of that in sight yet, it came down to either this lens of the 300mm f/4 IS L prime and the 1.4x tele-extender III that I own. The zoom proved to be much more flexible for the same cost and overall sharpness.
Hope this helps you out and gives you some solid info to decide. Above all, if you can, go try one out in a store and see how you feel about the reversed focus/zoom rings....as I said, if a new 100-400mm f/4-5.6 IS L comes out and does not have this dumb reversal, that would be the one reason I would sell this lens, however I would have to believe a new version of that lens would run about $2600 and not be available until 18 months after announcing it, given Canon's recent trends.