I've been using the 70-300L with a 5D mk.III for about a month now, and really like it. It is not only lighter than the 70-200 IS 2.8, but significantly shorter as well. It actually isn't that much longer than a 24-105L, though it is unusually thick. It telescopes out as you increase focus length, but there is a switch that will lock it at the 70mm position when you are carrying it around. It fits in the camera bag easily, and is just less of a hassle to carry around, whether on the camera or off. The extra reach is nice to have. Build quality is superb, of course.
The biggest disadvantage of the 70-300L to me is that the position of the zoom and manual focus rings is reversed compared to other zoom lenses. (Maybe there is some engineering reason why they had to do this to get such a wide focal range in such a sort lens?) On this lens, the focus ring is close to the body and the zoom ring is out near the end of the lens. I find this a little awkward, and it takes some getting used to, especially after changing lenses.
The other disadvantage is that it is a "Group E" lens, so although all 61 AF points are available on the 5D mk.III, only the ones in the center are cross-type. But the AF on the 5D mk.III is so good that I haven't had a problem with this lens, even for outdoor sports.
Yes, it would be nice if it were faster than f5.6, but as someone else pointed out, depth of field at long focal lengths gets pretty thin anyway. I think even if I had an f2.8, a lot of times I would stop down to 5.6 or more anyway, just to be able to get, for example, all the players in a group in focus at once.