Fallout 3. Of course, I don't have New Vegas, but my cousin does and I watched him play it and he let me play some... The only thing that kills me about Fallout 3 (other than the bugs that occasionally occur) is the fact that you have to have the Broken Steel add-on to play past the end.
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Music: Fallout 3 had more musical variety, but less radio stations. On Galaxy News Radio, you could hear oldies music, and on Enclave Radio, you hear patriotic music. On New Vegas, from what my cousin said, the music is all the same and it's "sucky."
Storyline: On Fo3, you find your father and purify (or poison) the waters of Virginia, while in New Vegas, you find the man who shot you in the head...? While both main quests are interesting, I believe that the quest for Fo3 takes the cake.
Main Character: In New Vegas, you don't know anything about the character, other than that s/he is a courier and was shot in the head by Benny. You don't really get to develop your character as much because you know less about them. In Fo3, you are playing through the major parts of your character's life since they came from the womb, such as birth, reading your first skill book, having your tenth birthday where the Pipboy3000 is received, taking the G.O.A.T test, and leaving the vault to find Dad. Therefore, there's much more character development.
People: The people in New Vegas aren't as food desperate, and it's almost like the nuclear war never happened. Besides, almost everyone is your friend and it's completely unrealistic. In Fo3, there are people (known as Raiders), that attack anyone on sight so they can get food. Some even take up residence in groceries, like Super Duper Mart. It's like...you don't know who's your friend and who isn't. 'Nuff said.
Weaponry: I will, admit, New Vegas beat Fo3 in this area. New Vegas has all kinds of mods and variety here where Fo3 does not. It also has more ammo (obviously) and from what I hear, you can craft medical supplies...? Correct me if I'm wrong on that.
Followers: Both games are great with this, but I do enjoy the little menu that comes up for followers on New Vegas. That's something Fo3 is lacking... It is a great improvement, but it's lame how every time you go to see Caesar, you have to abandon your follower before you can even go inside.