A full-frame sensor is physically larger than an APS-C sensor. APS-C sensors are 1.5X smaller linearly which is why they are also called 1.5X cropped sensors.
Full-frame sensors are known to have higher quality because they have bigger pixels. Bigger pixels means less noise and higher dynamic-range but of course there are variations. If you compare modern cameras, then full-frame models are indeed better in terms of image quality but if you were to compare older models you will find the latest APS-C sensors to be better than previous generations of full-frame cameras. Full-frame sensors also allow for a more shallow depth-of-field which is seen in classic portraits and abstract photography.
It is also important to know that a camera which has a bigger sensor requires bigger lenses. This means that you should expect to by buying bigger more expensive lenses for you bigger more expensive camera, assuming you compare the same grade of lenses of course.
The D600 has a rumored price as it has not been announced yet but the different in price will most likely be more than $400. If it does come in a a surprisingly low price, expect the NEX-7 and A77 to drop in price soon after.
There are two things to consider when comparing image quality, one is the lenses you use and the other is print size:
- The smaller you print, the less image quality differences show. A high megapixel count DSLR can make tack-sharp prints which are very large but you only print common sizes, then you wont be taking advantage of it.
- Lenses are one of the limiting factors of the system and unless you use top-quality lenses which are expensive and relatively heavy you will not see the full image quality your camera can record. So if you think you will just buy cheaper lenses after spending more on the body, that you will cripple your camera.
The crop-factor applies to the field-of-view of your lenses too. So a lens on an APS-C camera has a smaller angle-of-view, just like a longer zoom, than the same lens on a full-frame camera. This can be an advantage for shooting wildlife and other distant subjects by requiring smaller and lighter lenses.