DISCLAIMER: I'm not in the military, but I know and have talked to about a dozen people who have served in each branch. I can't speak about their mission too intelligently, but I can tell you what each person is generally like:
Samadma mostly hit the nail on the head, but here's what I've gathered from talking to both people in the Army and the USMC:
As Samadma emphasized, if you are in the USMC, YOU ARE A MARINE. Other than Spec-Ops (Ranger, Seal, etc.), Marines are the people who have a hard-on for getting in the middle of the fight. They are motivated by one thing: they want to kill bad guys. This can be a good thing and a bad thing.
It's a good thing because these guys are true warriors. If you talk to someone and they tell you they've served, you can tell rather easily whether or not they were a marine. I guess that's the "espirit de marine" or whatever. That distinction just isn't there for the other branches. According to those in other branches, they will hold a line better than most any other unit out there. There is no mission too dangerous for a marine.
The bad side is just that: the marines are often sent on the most dangerous missions, and they are the first to go in. A friend of our fam was an officer of a Ranger unit for around 15 years, and anytime he got a mission that expected high casualties he would defer it to the marines (or at least the initial attack).
There's a fine line between bravery and stupidity, and sometimes the USMC crosses it...and they do it with pride. Sometimes too much pride (The marines I've spoken to usually don't think too highly of people in other branches. I got all but laughed at when I walked into the recruiter office and a marine came out to ask if he could help me, and I said I was here to speak to Chief so-and-so.). But, they do their job extremely well and our military needs people like this. If you're the type of guy who craves a fight, by all means join the marines.
Because of this, the USMC has some of the highest physical standards next to Spec Ops. But, unlike Spec Ops units, the USMC must often deal with using outdated weaponry (although a few marines I've talked to would prefer an M14 to an M16...haven't heard a soldier make such a complaint, yet). The USMC also has to be able to take care of itself in battle -- it does not use the AF/Navy for air support, for example. So in that regard, there's more versatility. On the other hand, the USMC specializes in, as one Marine pilot told me, "shooting people and blowing **** up," so all of the support staff -- doctors, engineers, nurses, priests, and some transport duties are borrowed from the Navy.
The army is more specialized to ground forces. Yes, there are airborne units; yes, there are chopper units, but not as many. However, the support staff that the marines use the navy for are all handled by the army. So this goes back to the "what job are you looking for" thing. The army guys I've talked to seem much more level-headed than the marines I've talked to. Now, do they get that from the environment, or were they like that before they went in? I couldn't tell you. The motivation of soldiers is different from the motivation of a marine.
Now, others can speak about the mission, what each branch does, etc. much better than I could. But, to me, it's the people that truly set apart these organizations. So the real question is, what type of person are you?