This is a pretty strongly Canon-biased forum, so expect a fair bit of scepticism.
I was in a similar boat a couple of years ago; I wanted to step up to full frame and the choice was either 5D3, or D800. Everyone is right, the performance is essentially the same, and the button layout is different, however for me the colour rendition straight from camera is what drove me to Nikon.
Canon images, to me at least, are flat and just lack colour. I will certainly concede though that Canon's line of lenses is marginally superior depending on our usage. Another interesting point I've found is that when I was with Canon, I always wanted to upgrade to the latest model. I don't find that at all with my D800. I've had it for several years now and I wouldn't dream of selling it any time soon even though it has been superseded.
Interestingly, I've found a similar situation in mirrorless systems. The mirrorless king (Sony) I found to be too clinical and characterless, whereas the Fuji X series is just stunning in all aspects.
At the end of the day, it's a preference though, there is definitely no right or wrong answer.