I have both the 5D mk III and the Nikon D800. The Canon is great at low light, and is a perfect camera for gigs with its real silent shutter. The Nikon has great DR. End of. I shoot mainly landscapes, so I want good low ISO performance. The mk III forces me to use noise reduction at ISO 100 - 400. This is a terminal disease for me, so I don't use the mmiii for very much at all now. The mk III is dead in the water as far as I am concerned because of the noise banding in shadows.
You must have a bad 5DIII, for my landscapes it's been quite amazing and the appalling Nikon Live view effort puts me off any Nikon DSLR for landscape work. The D800 might have less banding and slightly more DR, but bracketing and digital blending is still required for high contrast imagery. If you are using NR on your 5DIII, then I would suggest your camera is out of spec, are using poor metering technique or you are rushing your landscape work. If you are pulling so much out of the shadows, then there is obviously a meeting issue or you are cutting courners with your bracketing and blending.
This image below, I combined the foreground and sky exposures into one image, I had to wait for the sun to kiss the foreground but the sun position was then wrong. So taking the two images created a better photo and one which looks balanced for exposure and has a stong visual feel. The difference between 30+ and 20+ mp is mute here and I get to utilise the camera's low 100 iso virtues because the 2 source images were taken using the camera's optimal performance.
I rarely have to use multiple shots with the D800 whereas I often had to with the mk III. In my experience I can also get more detail from the shadows without getting the noise banding. I agree that the Live View is far better on the Canon than on the D800, but then I very rarely use Live View in any case as I still prefer to use the viewfinder. I'll often shoot handheld which is something I do with my medium format 645D also.
My experience: Canon 5D mkIII Nikon D800
Live view better worse
Silent shooting much better hardly much quieter than normal shooting
Dynamic Range worse Much better
Low ISO noise much worse Much better
Shadow recovery much worse Much better
I'm no fanboi. I judge from my images. The Mk III can take some great images, but the pattern noise killed it for me. That being said, I haven't yet sold all my L glass, so I am keeping my head in. I operate a two camera set up: D800 and the 40 mp Pentax 645D. The resolution isn't the be all and end all. BUt is good to have the ability to print to about the same size with both. The D800 is the best of all three for dynamic range in my experience, but the 645D gives a different feeling. It also doesn't give me the pattern noise that the Canon does. And I'm not the only one reporting that.