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!
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, so you're being less demanding of the optics and can expect a sharper image for the same number of megapixels. It's true that some lenses get softer toward the image, but in general APS-C cameras are taking the middle out of the lens and blowing it up, losing sharpness in the process in a similar manner to a teleconverter. See this related question:
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.
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!
Lens hoods are designed for FF image circle and are therefore slightly more effective on FF cameras. This mostly applied to prime lenses. 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.
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.
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) - you can always counteract the shallower DOF that full frame gives by stopping down and relying on the better noise properties to up the ISO a little.