I'd take the 6D over the D600 because it fits my hand better, I already use D800s, and I'd prefer a replacement for my FF Canon disappointment known as a 5D2.
I shifted from all Canon bodies to a mix of Nikon & Canon over a year ago because I could not push Canon files to the extent I needed to without running into their low ISO patterned noise issues.
My first Nikon was a D5100, cheap, great IQ, fits my hand well and was easy to get used to. Tilted sensor issue affects most of the 8 I've now tested but... they're under $400 now! Best IQ/$ I've seen.
Next came a pair of D800/e bodies. I love the ergonomics and compared to my 5D2 the raw files are fabulous for what I need. There's a button for nearly everything and once used to it it's very much a photographers' tool, programmable and adaptable enough for most jobs. Requires 2 hands to operate more often than a Canon body.
I planned to sell one D800 and replace with cheaper D600 but, like the D7000, the ergonomics of the D600, for some reason, presses my irritation button, instantly. Just plain hate the feel of the D600 and D7000! It also was not worth the $1k savings to not have all the extra goodies I was starting to enjoy using on the D800.
I'm ready to toss my 5D2, it's raw files are too affected by low ISO pattern noise.
I had a quick feel of the 6D and like it. It feels like a decent enough layout and likely performs better than my 5D2 but need to make my test shots before I know if swapping my 5D2 for a 6D is an actual upgrade.
Also, with rumors of improved sensor tech coming from Canon, I'd like to see a FF body that actually performs really well at low ISO. I can't say, today, if the 6D's low ISO has improved enough to be comparable to the $380 D5100's raw files. 6D's hi iso performance certainly looks better but that's not what I really need to augment my kit.
I want a FF Canon that performs much better than the 5D2 for low ISO IQ at similar resolution, nothing else is needed.
I will not buy another Canon body, however, until it is essentially free of low ISO fixed pattern noise like current Nikons are.
Gotta give it to Canon tho, they make a better user interface in every body they make, very intuitive. Nikon's UI takes a lot more getting used to, often puzzling what some functions are until you learn them.
OTOH, Canon seems to re-arrange button layout on every model, so I often have to pull out a flashlight and look for a control in the dark. Kinda wish they'd pick a layout and stick to it for a while.
One more major thing for me is that Canon's live-view works a heckuva lot better than Nikon's in low light. This is 5D2's amazing feature in comparison to even the latest Nikons.
Happy New Gear!