As others said, speed is basically the biggest difference. Everything about the D4 (and any of the Dx series) is simply faster than any other model. Frame rate is the obvious one, but the built-in AF motor is also much stronger than lower-end models, for example.
In terms of bare specs, the D4 exceeds the D600 by a level or two in numerous areas, but the gap is relatively small when one compares it to everything the D600 is already capable of.
I have to dispute this. The gap is small only for your needs (I assume). Create in-depth comparisons and you'll find many differences that add up to two very different cameras. Under difficult circumstances, those differences are substantial and are what let you get a shot with the D4 and miss it with the D600. A big difference is the AF system: you won't realize how much better the D4's is until you use it in difficult situations. There's simply no substitute for shooting at ISO 204,600 with the D4 (as opposed to the D600's 25,600). If you need those capabilities the D600 is simply not able to do the job, and that's all there is to it.
More importantly, the minor features of the D4 are what really make it stand out, and are maybe even things that will one day trickle down to consumer-grade cameras. The feature of the D4 I most lust after is the illuminated controls -- all of the buttons are backlit to make them easy to see in the dark. By all accounts, the small joysticks are very fast and easy to use. The D4's self-diagnostic shutter mechanism likely helps attain that super-high 400k shutter life (compared to the d600's 150k). (So, in fact, you would need to buy almost three D600s to reach the typical shutter life of a D4 -- and hey, you would have spent the same amount of money at that point, too!)