While owning a 5DII and Rebel T3i I rented a 5DIII, 6D, D600 and D800e, have sold the first two and now own a 6D and 5DIII. I would have been quite happy to own both brands, but couldn't find any advantages in the D600 over the 6D that mattered to me (and while I certainly see the appeal of the D800e, I would probably have to change my shooting style to make the most of it, which seems the wrong way round).
It's surely undeniably true that the D600 has less shadow noise than the 6D at low ISOs (both are better than the 5DIII in that respect), but that only matters if you frequently underexpose or have dark shadows you want to brighten more than you can with a 6D. For me that's simply not an issue, but maybe it is for you. It's also true that the 6D is a bit better than the D600 at high ISOs, but the difference there is smaller, perhaps trivial. Otherwise, comparing photos taken by the two side-by-side (which I did only casually) I couldn't see any differences that I would call significant and that I could attribute to the camera rather than the lenses.
In terms of handling, I much prefer the 6D to the D600 ergonomically; changing settings seems simpler on the Canon, but as you're used to Nikon you may not agree. In some respects the D600 may have autofocus advantages over the 6D, but I found the 6D's focusing more accurate (though I guess that may be a lens issue rather than a body issue) and, in very low light, the 6D would focus accurately where the D600 simply wouldn't focus at all. I also didn't care for the slightly olive green cast to the D600's viewfinder or the green bias of its monitor.
I also very much like the quiet shutter modes of the 6D and 5DIII, especially in quiet places indoors (museums, churches, etc.).
Finally, for my purposes at least, Canon's range of lenses is better.
You could always rent one and find out first hand....
