I just got the 5D Mk III, as well as a D800 and D4, so my experience with them is limited to just the past few weeks. I also had a 5D Mk II, so my brief usage does let me compare them all.
The Mk III is a great improvement over the Mk II, which to my mind was like a Nikon D700 that lacked many of the D700's most useful features, including 42-fewer focus points, idiot bracketing limited to three shots, no built-in CLS-style flash, a less rugged body, etc. But the Mk II did have more megapixels and clearly superior video capabilities.
The Mk III solves many of those problems, particularly autofocus, bracketing, build quality and some others. Still no built-in flash, but when used with an external flash, especially the 600EX-RT, it gains more advanced CLS-style wireless control, both with optical and radio communication.
I don't see a direct comparison with the D800 which is aimed at a different user. The Canon is more of a general purpose camera with the limitations of such a camera. Despite its focus, the D800 *will* be used as a general purpose camera, and, given its 16MP resolution in DX mode, may actually have some advantages in the "utility infielder" arena.
The D800 does have features that especially appeal to those who actually do need higher pixel counts, at the cost of a surprisingly small amount of high ISO performance. The Canon is better at high ISO settings, but the D800 is not as "bad" as I expected it to be. I'm finding it quite usable at ISO 1600 or even higher.
I don't really see the 5D Mk III has being "better" at high ISO settings than my Nikon D4, however. Canon has always appeared to do more high ISO NR than Nikon, resulting in a plasticky appearance by default that many see as less noise. I see it more as a loss of detail caused by blurring the grain. I really like my Nikon D4's default noise processing, and while there doesn't seem to be an improvement over my dear, departed D3s, my results so far are good. I still haven't shot any concerts or indoor sports with it, and those will be the proof of the pudding.
Canon has some excellent lenses. They tend to do better in the telephoto range, especially in terms of variety. You can get a 70-200mm f/2.8 *and* a lower-cost 70-200mm f/4, for example. Nikon has tended to do better in wide angles, and I don't know if Canon will ever have a lens to equal the 14-24mm f/2.8 Nikkor. Canon has a very good selection of tilt-shift lenses if that's your thing.
Many people like Nikon's ergonomics better, but that's a personal choice. Most of the Nikon vs Canon debates usually boil down to a.) Switching camera lines is expensive and probably won't result in your taking better pictures. b.)How many lenses do you already own? c.)Does Canon/Nikon offer a model with an absolutely must-have feature, e.g. a particular video capability, or whatever? d.)Both companies make excellent products and will always be leap-frogging each other one way or another, so it's a moving target, especially in the digital age.