Lenses, Diffraction Limitation and Moiré
Resolution loss due to diffraction will start creeping into D600 shots around f/11. However, with the D7100, the onset of diffraction begins at f/8. I want to emphasis "begin", as resolution doesn't drop off after the initial point of diffraction limitation... it's more a case that contrast begins to gradually decrease. What this means is that moiré is only a potential problem for the D7100 below f/8. Most people don't expect the moiré to be a a big issue in everyday shooting, but my guess is that fashion photographers might have to deal with it (textiles and clothing) more so than others. Note that if you are a landscape aficionado, the removal of the anti-aliasing filter isn't that great of an advantage at the smaller apertures. Truth be told, as the megapixel counts have increased, cameras like the Nikon D7000 and D800 were already using lighter filtration than cameras in the past.
Diffraction-wise, the advantage goes to the D600. Within it's aperture range, you have a slightly wider latitude in aperture before diffraction starts setting in. Whatever advantages that the D7100 has in removing the low-pass filter are mitigated by the high pixel density of the sensor and the demands this places on the quality of the lenses being used. Even though full frame lenses are cheaper than DX lenses, the larger pixels of the D600 place less of a burden on the quality of the glass.
Speaking of lenses, the kit lens for the D7100 is still the 18-105VR. This lens has been maligned a fair bit; it's really not a bad lens as it's actually quite sharp, but it suffers from pincushion distortion and high levels of lateral chromatic aberration. That said, the 24-85VR is a better kit lens for the D600 than the 18-105VR is for the D7100. You can use the D600 combo without really noticing the lens flaws; not so with the 18-105VR.
Another advantage of the FX over DX is that for the equivalent depth of field, the FX camera will be using a lens one stop down. Imagine going for a trendy super-shallow depth of field shot with the D7100 and a 50mm f/1.4 lens. Wide open, past primes have all sorts of optical flaws, though some would use the term "dreaminess" rather than "flawed". For the equivalent D600 image, an 85mm (because a 75mm doesn't exist) lens shot at f/2 would give the equivalent depth of field. The D600 image would be superior, because stopped down, the FX lens would be operating in a more optimal portion of it's aperture range, meaning that even though both lenses would give similar looking out of focus backgrounds, the D600 and FX lens would be more contrasty in the center of the image, presumably where the subject is.
Safe Hand Holding Ability
Assuming that the D7100 sensor is a variant of the D5100 sensor, my guess is that under non-professional usage ISO 3200 is about as high as you would want to take it. ISO 6400 would by my limit with the D600, which is in keeping with the 2.3x increase in sensor surface area. Under equivalent circumstances, you would get one additional stop in terms of safe hand holding speed. However, one additional stop is not the advantage that it used to be. In the D3/D300 generation, ISO 1600 was as high as you would want to take DX, and ISO 3200 for FX. As the ISO ranges of cameras increase, the additional stop of light gathering power becomes less significant.
With the current set of cameras, assuming a viewing angle the equivalent of 50mm FX/30mm DX, you can safely handhold the D7100 at maximum ISO and aperture in an indoor room. The extra stop that D600 gives you means that you can safely handhold if the room is dimly lit. Truth be told, I wouldn't push it this far. First, I'm excluding t-stop calculations for the efficiency of the lenses used, so the real world values would require more exposure compensation to achieve proper exposure. Secondly, under these circumstances, I would have long ago turned on the camera flash.
Outside of stationary objects, speed is speed, and the fact that D600 can go high up the ISO range means that it is more suited for capturing fast moving objects. However, what FX gives, it takes away, so if you've been using a lens like the 70-300VR, it now becomes "just" a 300mm max lens, whereas on the D7100 it would be the equivalent of 450mm... 600mm at 15mp in the D7100's high speed crop mode. Combine this with the D7100's 51-point autofocus module and you can make a case that the DX camera is actually the more versatile of the two.
The lack of an anti-aliasing filter shouldn't be a problem with the D7100. Many still cameras are prone to aliasing artifacts in video, but this is due to the fact that not all of the pixels are being used, and the ones that are being used are spread across the sensor, not adjacent to one another. One frame of 1080p is roughly the equivalent of a 2 megapixel image; in other words, a modern dSLR sensor has more pixels than what a high definition video stream needs. To get around this, only a certain number of pixels are used via sub-addressing. In other words, if aliasing artifacts are an issue with the D7100, they will be because of the reasons that plague all dSLR video functions, not because of the lack of the low-pass filter.
It looks like the D7100 has inherited the video functionality of the D600, so my guess is that for the most part, the D600 will have video that works in almost the same way. Naturally, the D600 has larger photosites, so you will have more exposure option and depth of field options. The D7100 has mercifully moved the microphone to the top of the camera, away from the autofocus motors. If you were to do serious videography, you would use a proper external microphone, but for casual usage, I would prefer the D7100 over the D600.
Value for Money
This one's not a contest, the D7100 gives much more bang for the buck than the D600. Both cameras can do most things on a relatively equal footing. The D600 does what traditional full frame cameras do well: creating quality images that have a smooth tonal transitions. There is no question that the D600 will produce higher quality and more pleasing images, but you have to have the ability to bring it out of the camera if you want to produce images that stand out from the DX crowd. Think of all of those "what camera did this come from" threads that you see on DPReview... The D7100 is a more versatile camera, though, and its high speed crop mode and more advanced autofocus system alone make it a more intriguing choice. That said, both models represent Nikon well at their respective price brackets, and offer photographic capability that was undreamt of at these prices not just a few years ago.