Hi, the 70-200 + 5D3 is a killer combo but here is my opinion:
I have both the 70-200 2.8 II and the 70-300L + the 5D3. I rarely take the 70-200 with me if i'm going on a vacation or a trip. It is just too heavy for hiking/carrying around all day. The 70-300L is much better in this regard and as a bonus it's longer. If you add a 24-105 or 24-70 you won't need anything else for general purpose shooting.
The image quality is of course a bit better on the 70-200 (especially when stopped down to f/5.6 or f/8) but mainly noticeable in the corners. For situations other than very dark dawn or dusk or pixel peeping test chart shots the 70-300L will deliver quite good enough image quality compared to the 70-200 and does so in a much more travel friendly package. The IS and AF are also on the same level in my opinion. You can also ad a 1.4x Kenko TC but I've found almost no difference just cropping the image.
Then again the 70-200 will be used exclusively when I'm shooting an event, a wedding, sports indoors or portraits. But these will be things I will go to shoot for one or a couple of days as the main thing and not a holiday or a trip.
So in my opinion it depends on your shooting needs. If you plan to use the lens as a general purpose tele on vacations and trips then go for the 70-300L and maybe buy a 50/85 f/1.8 or a macro to accompany it. But if you must have the best IQ in a zoom or need the low light cabapilities for event/indoor shooting and are prepared to carry the weight around if you occasionally use it as a general purpose lens then go for the 70-200.
Or you can do what I did and end up getting both