I shoot more landscape than anything else, although animals, birds, sports, and motorcycle racing are also of interest.
At f/8, ISO 100 from a tripod,it's going to be hard to tell APS-C from FF. If you need to crank up the ISO for a fast shutter, FF will have an advantage. If you need to track a fast-moving subject, the 70D/7D AF will be an advantage over the 6D. If you need both high ISO and AF tracking (which you often do for animals, birds, sports, and motorcycle racing), the 5DIII delivers.
I'd agree with all that, though I will say if you plan on using EF L lenses, then a 17-40 (or 16-35)will be a 27-64 on a full frame. And if you need something really wide because you can't walk backwards, full frame will deliver. Though the Tokina 11-16 works really well from what I've been told, so there are options.
I think you could do sports and motorcycles using the center point of the 6D, but if you try movement on the other AF points, it won't do nearly as well. Obviously, you would adjust your composition of the shot in post.
This also has been beat to death, but your lenses will usually make more of an improvement than a new body (usually), so it depends on what you have no and what you are willing to part with.