You will find the control layout so similar, that the transition is painless. My only gripe to this point is that the L-brackets for the D7000 do not fit the D600. Identical memory cards, batteries, AC adapters, wired and wireless releases, chargers and eyecups are a pleasure.
The difference in AF sensitivity was a big selling point for me. The promo brochure on the Nikon Support site for the D7000 states that the D7000 9 cross-type sensors work with all AF NIKKOR lenses f/5.6 or faster. However, the Nikon brochure for the D600 states that If the maximum effective aperture of your super-telephoto lens gets as slow as f/8 when attaching a 2.0x teleconverter, you can still count on the D600 AF performance with seven active AF points.
This was first brought to my attention in the D800. The ability to get useful AF function out of the 1.7x and 2x converters is high on my list of desires. My testing revealed impaired AF when these are mounted to a f4 lens on the D7000. The 1.4x converter is OK. With the improved sensitivity on the D600, the 1.7x should become as good as the 1.4x was on the D7000, and the 1.4x should work very well indeed. I think the 2x usefulness is just more marketing hype, but I need to test to make sure. I have high hopes for the 1.7x. Also, this would make the newly announced 800mm f5.6 useful with the 1.4x converter.
EDIT: Testing on the 600mm f4 shows me that the 2x converter should remain on the shelf. The 1.7x is now well within the OK range of functionality, and the 1.4x works great. If you use TC units on long lenses, the D600 (also D800, D4)beat out the AF performance of the same combo on the D7000.