[mst]

Why don't you go to a dealer, grab one of each and see which feels better for you?

That's the best advice one can give to such a question. Grab both, see which you like better. Shoot with both, and see which you can manage better.

Decide after that. And from there on continue to think about which lenses or other accessories you may need.

Both the D600 and the 6D are very capable cameras.

-- Markus