My advice would be that you bring a cash deposit to a store along with a memory card of your own so that you can borrow each of the cameras one at the time to shoot for half an hour outside in the street in front of the store.
Deliver back each camera and get your money back and go home and evaluate the image rendering of the available light captures on your memory card.
For some mysterious reason most photographers tend to prefer to make
statements over showing comparison
illustrations even when they own both relevant brands or both relevant formats for a given discussion.
I never learn to understand this but that is how it is.
By this I do not mean to say that statements are totally useless, because at least the statements do tell you something about what to look for when you try out the gear yourself.
Fred Miranda compared a couple of other models (5DIII & D800) of the two brands you mention, not directly relevant for your choice but still it might give you a hint about some things to look for (like the rather poor implemented LiveView on D800 which I believe is the same on D600).
You can read Fred Miranda's comparison
here.
(P.S.: There's also this
6D - D600 comparison on YouTube, haven't seen it yet myself)