New Vegas opens up much faster. It doesn't take time for exposition. You wake, some doctor tells you got shot in the head and whatever and lets make a character and have some fun.
As some guy called William Shakesman once said: "Brevity is the soul of wit."
It really is amazing how slow and boring the introductions are in
Fallout 3 and
Fallout 4. Fallout, Fallout 2 and
Fallout New Vegas all give you a quick intro, a decent little set-up, then BAM, get to questing. By the time you've gotten out of your stupid vaults in both
Fallout 3 and
4, chances are you've already done a number of Shady Sands quests in
Fallout.
It should also be noted that Bethesda's story places a sense of urgency on a setting not suited to urgency. GOTTA FIND DAD and GOTTA FIND SON are things most people wouldn't take lightly, but the worlds are so distracting a lot of the time it's very easy to forget about the main quests. Same goes for
Oblivion and
Skyrim. What's that? Dragon destroying the world? Yeah hang on a sec I'm just helping this guy find his missing scroll in a cave somewhere.
While
Fallout gave you a timer, because the quest given to you actually is urgent, both
Fallout 2 and
Fallout New Vegas are designed with the possibility of playing a character who could give less of a shit about the main quest.
New Vegas gives you a vague idea of revenge against the guy who shot you, but honestly, you'd probably just want to go to Vegas anyway because you can see it for miles and it looks more interesting than rocks. The worlds are designed for moseying around, the stories complimenting that ability to roleplay.