It's a given that New Vegas plays better than Fallout 3 because it took the game's engine, perk system, among other things, and improved on it. With that being said, I feel as if the "retro-fifties/forties" theme that is widely illustrated in Fallout 3 is lost in the wild-west theme of Fallout New Vegas.
Story? No competition. New Vegas is *far* better in that department. Exploration? Eh..... Both have their ups and downs. Fallout 3 has a bigger map with more meaningful/open random areas, but a lot of the locations are recycled over and over again. New Vegas tends to make its locations quite unique while a lot of the areas are simply boarded up and are of no relevance.
Weapons? Perks? Again, no competition. New Vegas has a wider array of personal customization of your character's play-style in terms of weapons, varying ammo types, weapon-mods, and perk selection.
DLCS? Hate to say this, but again, this is not even a competition. New Vegas' DLCs were all connected, immersive, rewarding, and difficult, while Fallout 3's were (besides Point Lookout) mediocre IMHO. Zeta and Broken Steel were awful, while Anchorage was mediocre and the Pitt was decent.