I had the pleasure of using the D600 & D7100 alongside each other.
It was a tough call, but in the end I went with the D7100 and have never looked back.
For me the D600 fell short of expectations and the D7100 just done what I wanted it to do better.
Build quality they are pretty much identical, although I did find the battery door and few other covers better on the D7100.
The D600 focus area is very small - if you shoot subjects at the side of the frame you end up doing a lot of focus and re-compose.
Speed wise - D7100 shoots more FPS, focus speed seemed pretty much identical for both.
Controls - pretty much identical.
Size and weight - D600 slightly heavier and bigger, not by much.
IQ - for me there was nothing in it up to iso 3200. D600 was better above 3200, but nothing to shot about.
The removal of the AA filter in the D7100 an improved sharpness a lot from the D7000 IMO.
If you want full frame, by all means choose the D600. Its a great starter FX camera, just don't expect it to blow you away like a D800 would.
Finally - Both the D7100 and D600 need decent glass for the best results. I'm not saying full on f2.8 glass, but a step up from the kit lenses.
The D7100 for me, is the best crop sensor camera to date. I just cant complain and the quality of images it produces.
The D600 was made for a cheaper way to FX, but I think Nikon could have improved it. The D7100 has the latest 51 point AF system, the D600 had the older 39 point focus system.
Its not a bad camera by any means, but I think the D7100 is a better camera and thats why I chose it!