There's no meaningful difference in image quality between the 6D and the 5DIII. For all intense porpoises, you can think of them as being loaded with the same "film."
But the bodies holding that film are quite different.
The 6D is a capable camera, but it's nothing to get excited about. (Except, as already noted, for the sensor.)
A decade ago, the 5DIII would have been the reigning champion amongst sports photographers. In the film era, there would have been no contest.
Sure, its frame rate isn't quite as high as some, but the autofocus performance more than makes up for it. Except for the 1Dx, you get more "keepers" per second of shooting than with any other camera.
That, and the price, is really all you need to know about when deciding which to get.
Basically, the 5DIII is a pro sports / wildlife / action version of the 6D.