Conclusions
In terms of image quality, the Nikon D800 is the winner: after post processing, you get images with more detail and slightly less noise than 5D3 and 5D2. Overall, the difference in terms of noise between the three cameras are minimal. This is an impressive result for the D800, considering that it has much smaller pixels than 5D3 and 5D2.
The image quality of the Canon 5D Mark III is good, but it is disappointing if you compare it with the previous 5D2: considering that three years have passed and that the 5D3 is much more expensive than 5D2, I expected at least 1.5 or 2 stop less noise. Instead, there are no real differences up to ISO 1600, and from ISO 3200 to 25600 the 5D3 shows a little of advantage, but it is something near 0.3-0.5 stop, that is much much less than expected.
For landscape, macro, still life, portrait and other "static" genres of photography, I don't see many reasons to upgrade from 5D2 to 5D3, while if you are a Nikon user the D800 is a fantastic upgrade from the D700 - the jump from 12 to 36 megapixels is huge to say the least.
If, instead, you take sport or wildlife photos and you use the 5D2, you may consider the 5D3: even though the image quality is about the same, the 5D3 is faster than its predecessor; AF and continuous shooting have been improved. If you use Nikon, you have to trade a lot of speed to get the great image quality of the D800, so for some users the D700 may still be a better choice.
In conclusion, both cameras have their pros and cons. The 5D3 is a better all around camera, while the D800 is the winner for the "static" genres of photography: you have to choose the one that best fits your necessities. Personally, I think that the perfect camera would have been a 5D3 with the sensor of the 1DX: similar resolution to the 5D2 but a truly improved high ISO performance.