I have both for the time being. I got the 600mm first because I thought everything was shifting to full frame and was looking to replace an 800mm. Like others have said, for birding, I think the 600mm II + 1.4x III is the best option. If the birds are stationary, then the 600mm II + 2x III will also work nicely (super sharp). This is not a lens you want to travel that far with though, since it is big volume wise. Also hiking around with it is cumbersome at least for me. So that is something to consider. I mainly use the 600mm II on a sturdy tripod and a full gimbal head.
The 500mm is lighter on paper, but in my hands it does not feel that much lighter. It is however, much smaller in volume. So would be much easier to take on a plain or even to hike around with. Still not sure I want to hike around with it much. If you think you might be sticking with a crop sensor body (7D, 70D and maybe the next 1.6x body) then I might suggest you strongly consider the 500mm. If not for the slight weight savings and size differences, then for the price difference. The 500mm + 1.4x on a 7D is more than enough focal length for birding.
I also have the 300mm f/2.8L IS II and would choose to use that, if I am going to be hiking around looking for subjects to photograph. I'd use it without hesitation with the 2x III. It is much lighter than either the 500mm and 600mm and it is super sharp even with the 2x III on it.
Don't know, if that helps you any. You might even consider a nice used 800mm f/5.6L IS that can be had for around $8500 these days.
Best of luck with your decision. If you decide on a 500mm II, I might have one for you :-).
Thanks Jason, I would readily buy a used lens but given the fact that I'm in India, it is not an option. The superteles don't come up for resale all that many times. Buying from the US isn't such a great idea because imports attract a customs duty of 30%. A new super tell lens, in India, turns out to be the same if not cheaper.
Carrying the lens in the wild... I'd take a couple of sherpas for that instead of breaking my back with either supertele.
I'll go for the 600 next. The 300 should be here within a week.