Early model D600s had an issue with the sensor getting oil splattered on it by the shutter. Later model D600s as well as refurbished D600s have a newer shutter, which is the same as the D610. There are no oil splatter issues with the new shutter.
If you can be sure you are getting a D600 with the newer shutter, then it would be pretty much the same as the D610. But the risk is getting a D600 with the oil issue (the D610 has no such issues).
I have a friend with a D600, and his oil issues continue. Some people have reported that after 3,000 or so shots, the oil issue dissipates, but his camera has many more than 3,000 shots on it, yet it still spits oil on the sensor.
On the other hand, he has become a pretty good expert at cleaning sensors.
The D7100 is a good alternative (I own this camera). While the D600/D610 does enjoy a few advantages over the D7100, they are not nothing a beginner would notice. You get a bit more low level performance and a very minor improvement in image quality due to the D600/D610's full frame sensor vs. the D7100's cropped sensor, but the differences are really not significant.
But the true cost of ownership for the D600/D610 may be more costly more if you consider lenses, as you have to buy FX lenses for the D600/D610. You can buy FX or DX lenses for the D7100.
I ended up buying the D7100 over the D600 in May 2013 (the D610 was not yet available), as the images were a bit sharper due to the removal of the aa filter on the D7100. So it basically came down to sharpness (D7100) vs. low light capability (D600). And when I added up the lighter camera (D7100), and lower cost (D7100), I ended up choosing the D7100.
Even today, I would probably chose the D7100 over the D610, simply because out of the 13 Nikon lenses I have, 6 of them are DX lenses, so I would have had to buy a few new lenses. That is what the cost of ownership meant to me, and it would have been far costlier to replace the 6 DX lenses than the cost difference of the D600/D610 vs the D7100.