So much to the myth, that Nikon users are happier with their gear.
And the grass is always greener on the other side.
Fact is, that Nikon has the better Camera, on paper that is at least. 36MP for less money, and a much higher rating at DXO Mark.
The actuall question is, will you take better pictures with the D800 than you do with the 5DIII. Probably not; and also not the other way round. There are differences between the two cameras, some of them are easely measurable, others not.
To me this is like boys comparing cars. The BMW M3 has so much HP and this 0-60 time, but the Audi RS4 is stronger and even faster, and it has 4-wheel drive. All true, but put them on a race track, and there you will see the real difference in the performance of the cars.
And the biggest difference will be, when you switch the driver, because both of them are amazing cars, who offer more than you actually need in most situations. The same goes for the D800 and the 5DIII.
What influences the quality of your pictures the most, are your skills, and I think there are not so many people who are so skilled, that they can blame their camera for bad shots. We can reproduce known issues, and we can see noise at high ISO, but what we have to learn are the limitations of our tools, which defines the way we are using it.
To go back to the car comparison, we will never be able to go over a certain speed in a certain curve, it's simple physics, you will only be able to reach the maximum speed given the car and the circumstances. If you are too fast and fly of, than it was your mistake, and not the one of the car.
So how big is the difference between the D800 and the 5DIII, and how bad is the 5DIII? I would say both cameras represent the crown of digital FF photography at the moment; what you think is better mainly depends on what is more important to you, or what like better. It is like with the M3 and the RS4, none of them is realy better than the other, they are just a little different.