- One major difference is that a FF camera produces a depth of field that's around 1.3 stops shallower than an APS-C camera for the same subject & framing. This is most important when you have the aperture as wide as possible, e.g. for portraiture. To replicate the look of a 50 f/1.4 lens you'd have to use something like a 31 f/0.9 lens, which doesn't as far as I know exist!
Quick and dirty comparison image, APS-C Canon 30D left, FF Canon 5D right, same lens (FF image was zoomed in, however to give the same field of view), same composition, both f/2.8
Another difference is that if you're using a lens designed for a full frame camera (like all Canon EF lenses) you are making full use of the image circle, which is less demanding of the optics and so you can expect a sharper image for the same number of megapixels. It's true that some lenses get softer toward the edge of the image, but you will still get higher average sharpness with most lenses, and telephotos will be sharper right across the frame. The crop factor of APS-C cameras takes the middle out of the lens and blows it up, losing sharpness in the process in a similar manner to a teleconverter.
Larger formats allow for sharper optics. One of the driving forces for larger formats (other than the relatively constant resolving power per unit area of film) has been that it allows lenses to be produced which resolve a greater number of line pairs per picture height. Going full frame on a DSLR benefits from this to an extend - see: With all other things equal, in a DSLR, will a larger sensor produce a sharper image?
A bigger sensor means bigger pixels, which in turn means you capture more light usually achieving lower noise levels in the process. Greater dynamic range goes hand in hand with this.
You get a larger, brighter viewfinder on a full frame camera, which can be helpful composing shots. Having said that, I personally find the 5D viewfinder too large, I've not used a 7D but it has a very high spec 'finder.
You have more mirror to move on a full frame camera. The larger heavier mirror usually means shooting speed is limited. The mirror on my 5D moves so slowly I can actually see the world slide sideways/up for an instant!
Likewise the mirror box, focussing screen and pentaprism are larger, meaning the camera is larger and heavier.
Lens hoods are designed for FF image circle and are therefore slightly more effective on FF cameras. This mostly applies to prime lenses, as zoom lens hoods are designed cut to accommodate the widest zoom setting, so everything else is already non optimal. If you're using an EF lens on a crop camera you ideally want the hood tighter (since the extra shading will lie outside the smaller sensor a tighter hood wont vignette.
I have nothing against APS-C cameras but for any format it makes sense to use lenses designed for your sensor size. The range of EF-s lenses is smaller than the range of EF lenses. However for some uses (sports etc.) the smaller sensor size is helpful for the extra reach and speed it allows. Also the better noise characteristics of a FF sensor don't quite make up for the higher ISO you need to use get the same exposure when stopping down to match the DOF as a crop. So if you have to maximise DOF crop has a slight edge.
If there are EF-s lenses available for what you want to shoot then it wont be noticeably worse choosing this camera. However I feel full frame gives you more flexibility (speed aside) - as you can get the same deep DOF as a crop, but go narrower if you need to.