Fallout: New Vegas offers four different "main storylines". That is to say, there is a war coming and there are four different sides to fight for. Each of the routes offers a different perspective on the factions, the people, and the area. You'll meet different people in each, and your friends on one side would be your enemies on the other.
Fallout 3 handles things differently. Here, there is also a war going on, but there are only two factions involved. You don't have a say in which side you choose, as you're forced to fight for the 'white knights' faction, against the bad guys. Fallout 3 offers little variety in its storyline, and is more black-and-white while New Vegas offers many different shades of gray to choose from.
However, while the focus in New Vegas is more on the story, FO3 is more focused on exploring. I have done many, many FO3 playthroughs and have played for countless hours. However, when I start up the game now, and play for some time, it's very likely I'll run into something or someone I've never seen before. FO3 offers many places to explore, and things to discover.
Last major difference in my opinion: Fallout 3 is dark. You have to understand, Fallout 3 is set in Washington DC, and that city got hit
hard in the Great War (huge nuclear war 200 years ago, turned most of the earth to ashes and continues to poison people with radiation 200 years later). DC was hit hard, and the city's mostly rubble. There's a lot of radiation, a lot of messed up, mutated creatures and the atmosphere overall is a bit dark and depressed.
Fallout: New Vegas is, obviously, set in Las Vegas, and it offers an entirely different atmosphere. The city wasn't hit by the bombs directly. There's little radiation compared to DC, things are more civilized and the overall feeling of the game is more colorful and upbeat.
That's not so say New Vegas is more fun to be than DC, it's a matter of personal preference. I love FO3 and DC, but the New Vegas atmosphere suits me better.
Which one to choose? It's up to you. FO3 is an older game than NV, so many gameplay and graphics issues were solved in NV, so from a gameplay point of view you might like New Vegas better. However, if you decide to play them both I'd recommend playing FO3 first. FO3 is a step back in areas like gameplay, combat, graphics etcetera, so you'd better play that one first if you end up buying them both.
Overall, I liked New Vegas better. The story, the characters and the atmosphere are better than FO3's in my opinion. However, I played FO3 before I played NV, and I wouldn't have wanted to miss it. FO3 has some great characters (Butch, Sergeant RL-3, Clover, Desmond, Eden), some great DLC's (The Pitt, Point Lookout) and is a very fun game that you'll play for hours on end.