WITH TWO ABOVE + A BIT MORE...
First, search the forum and ppl talk about it all the darn time, this Q is probably is as common as Canon vs Nikon.
Now the fun part, besides computer chip that's in the camera and w/e processing does occur, image sensor SIZE is compared to 35mm camera. Since I don't have actual numbers in front of me and don't feel like googling or opening my photo books to give you number I'll rely on you to look up 1.5x (most common Nikon crop), 1.6x and 1.3x Canon crop image sensors compared to size of 35mm and lovely
medium format size (not talking about
large). Just for references of sizes, here's a nice image from wiki
File:35mm MF LF Comparison.jpg - Wikipedia, the free encyclopedia
So if you're coming from 35mm film to cropped digital, first you have to remember is that what you're used to see in your view finder you won't see on crop. The popular nifty 50, the so called
normal lens, is only normal on croped sensor.
(Experiment - w/ one eye look through viewfinder and keep other open, play with zoom until you'll see no magnification - that is normal - what you see is what you get).
Until about 3-4 yrs ago, noise was an issue, rather bigger issue, with dSLRs, something that today it seems that it's a holy grail of new cameras (shoot at high ISOs w/ low noise). Advantage of full frame is that b/c the image size is bigger, it'll capture more 411 and (don't ask me for technical explanation

) will provide you cleaner, less noiseir results. Now, I'd tell you, I worked with Nikon D3 few weeks ago, and that sucker was cranked to ISO 2000, and compating images to my d90 and d300, difference was pretty obvious. (Yes, I mentioned Nikon, but Canons are the same thing). Disadvantges, its probably the only one I can think of, is that they are more expensive.
So back to crop and lenses... If my 16mm fish eye was a FISH EYE on film body and will also be a one on FF, it won't be on croped, thus I'll need a wider angle lens - I'll need to invest more $ into glass.
Today, with technology being where it is, the disadvantage that I could think of crop is noise at high ISO and
EVEN THAT IS DEBATABLE b/c it is a subjective topic to many. And wide angle/fish eye you need something of 10mm

. An advantage, is 1) crop is cheaper by on average by 25% AND 2) the nice portrait 150mm from film/ff is now about 200mm (based on which camera it's on) - you can use it for portraits and practice on deers until you get a 400mm or 600mm

.
DOF: b/c of the sensor size it also varies (as Derrel pointed out). The lovely f/8 on FF is about f/6.3 on Dx. Now for landscape guys it probably isn't a big deal (not sure though if shooting in the evening and lower light conditions it might be). For wedding photogs who know the power outage of their strobes just by the POP it produced when flash is set of, these 2/3 of a stop could be a hassle if want to achieve same dof as did with 35mm - not saying it is a hassle, but could be.
Earlier, I mentioned medium format. I believe,
but don't quote me on it 
, that until about 2-3 yrs ago, there were no dslrs who could provide image quality similar to medium format film. For both digital and film it is very important to remember how the image is shot and how it is processed. But today, technology changed and though I haven't seen any comparison (would have loved to though) but I think that news dSLRs are coming close to it.
Just for references, here's today's
Hassy film and compare it to
Hassy Digital. There are photogs out there who will spend 20grand on digital camera, for what purpose?-I don't know but if they have the money and willing to spend - GO FOR IT!!!
I hope this somehow shines a bit more light (by 1 or 2 stops of light

)on the whole sensor size issue. There really is no WRITTEN-IN-STONE ONE BETTER THEN THE OTHER, rather there are advantages and disadvantages of Full Frame and cropped framed.
EDIT: just a quick thing on megapixels. Most ppl don't print anything more then 4x6, and sometimes 8x10. Hardly any consumer today print even 4x5

. On my wall, I have a 16x20 hanging that was shot with D70s. Thus the common thing you might here people say that you can't print image size like that from 6megapix is INACURATE. Keep a lot of these things in mind as you grow more into photography and learn more about it.