Fallout 2 is basically Fallout 1, but the Wild Wasteland trait was included and made a mandatory option that you can't turn off.
I wonder what the pitch for Fallout 2 was.
"So... what if, now hear me out here, what if, we took a bunch of random shit and threw it in a blender and made a Fallout game out of it? I'm talking cowboys and the wild west, mafioso gangsters, stormtroopers, government conspiracies, vaults are a conspiracy too now, the entirety of San Francisco is basically just the movie Big Trouble In Little China, and - and get this: talking. Deathclaws. Awwww yeah Fallout just got a little bit more WaCkY team."
For the OP, if you haven't played it, Fallout 2 is technically more similar to the newer Bethesda and Obsidian games than the first one, but it's a completely different tone. Well, using the singular version of the word 'tone' would be incorrect. It would be more proper to say 'tones', because it feels like the creative team wanted to go into multiple different directions, couldn't decide which and then just said "fuck it, let's go in ALL the directions". Almost every town and faction in the game, especially mid-late game, has some kind of ridiculous over the top theme to it. Reno is basically Godfather-esque Italian mafia, Shi are outlandishly caricaturized Chinese-Americans (specifically Chinatown and Triad culture), et cetera.
Fallout 1 is much more serious, somber, humble, and while it has some bizarre elements or things that lend to it's retro 50's heritage such as 1950's styled energy weapons, Fallout 1 is the much more down to earth and "serious" nuclear apocalyptic game of the two.
In fact, I'd even go as far as to say that Fallout 2 having so much wacky bullshit in it is/was what directly opened the door for Bethesda to take the IP in the direction they have...