I'm a hobbiest and own both. I'd have to day that the 70-300 is sharper than the 70-200, though not by a substantial margin. That, along with the weight, make it my first choice if I'm able to use it. Its sharp enough to crop photos substantially for more zoom. Phenomenal lens, hands down (read the review of it in Africa on this site).
However, it falls short for low light. If you are shooting stills on a tripod, it can be okay, but still has a harder time auto-focusing (compared both shooting colorado wildfires this summer at night). Night sports outdoors (tried a rodeo), indoor kids programs, etc. - the 70-200 shines. The auto-focus is definitely faster, even in good light. So, you might find the 70-300L frustrating at a soccer game even at high noon.
One other option to consider... maybe pick up a used 70-200 2.8 non-IS and a 70-300L for about the same a a bit more money?
FWIW, I've been shooting these lenses with a 60D. With a FF, you may really want the extra reach. The clean high-ISO of the Mark III might make the 70-300L work for you.