I've got both a D800E and D600. I used the D600 for 5 months before recently converting it to IR (which was the original intent).
The D800 and D800E are very similar and for the sake of discussion here, I'll assume they are the same.
The D600 is a little smaller than the D800 but still fits well in my medium large hands. The D700/D7100 is very close to the same size but I find it to be small.
Image quality is very good on both the D800 and D600. Sensor performance is excellent on both and would not be a significant point of differentiation. The D800 has a larger sensor and can be used for additional flexibility in editing or for larger prints.
The other AF difference is the layout of the sensors. The 12 sensors missing from the D600 are in the corners, and on a full frame this can make a difference. Not only can this be with composed images, but if you are using AF tracking and letting the focus sensor change, you could run into an issue with the smaller number of sensors.
I use the "back button AF" on both cameras so I can separate focus from shutter release. That lets me use the more accurate center sensor a high percentage of the time.
In terms of weather sealing, I don't see much difference. Both are good but neither will withstand a dunking. Rubber buttons and flaps provide protection, but you do need to use proper care.
There are some advanced features on the D800 not found on the D600. The D600 has limited bracketing, limited multiple exposures (to 3 frames), and some other functions that are below the D800. These are functions I use from time to time but not a big deal for most people.
Bottom line, I think most people would be perfectly happy with a D600. Its a remarkable camera and a good value. If the D800 was not on the market, I would not have any question about whether the D600 was adequate. For kids and casual shooting, the smaller file size and faster frame rates are nice positives.