Full frame is sharper, has more dynamic range, less noise, and over a stop less depth of field. But unless you print fairly big, at least A4 or larger, you will be very hard pressed to see the difference except in terms of DoF.
It's not just about pixel count - full frame is about 2.5x larger than crop. Pixels are much bigger. Lenses don't have to work so hard. If you compare say a 7D frame with a 5D2, with roughly similar number of pixels, the 5D2 looks much better. The visual difference is much more than say the difference between a 40D and a 7D, even though the latter has 80% more pixels because they are so damn small.
The advantage for landscapes is obvious I think. For portraits, I would like to achieve shallower depth of field occasionally without resorting to very low f/numbers which really need an ND filter with studio flash. More practically, I often find I want to crop portraits quite heavily which pushes my crop format 40D right to the limit. For example, I try to shoot as tight as I can, obviously, but then when I review the pics (not that obvious at the time of shooting) I can see a better image if I crop in hard. Another situation that sometimes occurs is when shooting two kids at once, which is often unpredictable, and I find that I've got a really good shot of one of them that I want as a solo portrait. I find that I can only get a really good result out of such a huge crop if I make a smaller print than I would like. So being able to pull out just half the frame would be really handy sometimes.
Full frame has less reach though. Crop format wins with long telephotos for sure, if only because the equivalent field of view requires such absolutely huge lenses on full frame. A 500mm lens on a crop is just about manageable, whereas the equivalent 800mm lens on full frame is neither practical nor affordable. If you can do it though, quality must be just incredible.
I'm constantly reviewing full frame vs crop in my own mind. I elected to go crop, finally, about a year ago and I still think it's the right decision for me. A top end crop camera can do anything pretty well, and is cheaper and smaller/lighter (or at least the lenses are). Full frame however will just not do long lens birding anything like so well.
Hmm

That's got me thinking again. I go birding about four times a year...