I'll add my 2 cents:

Engine and mechanics-wise Fallout and its sequel are identical. There are a few odds and ends added for convenience, such as the ability to move more than 999 items of the same kind at once (useful with money), and a neat system for controlling your teammates' behaviour, as in "when should they run away" or the probability that they would shoot a gun in full auto mode.

The world was expanded both in scale and the sheer volume of quests and content. There are more weapons, armors and items this time around, and the world is bigger and more populated. Your stomping grounds are exactly north of where you treaded on the first game, so there are are few locales that you can revisit.
Also, there's a neat variety of NPCs that can join your party, more than in Fallout, and the selection includes Robots, Mutants, Wastelanders and a sentient bipedal reptile, among others. The NPCs in Fallout 1 were few and far between, and except for the dog, pretty generic in looks and aptitude.

I have no gripes about this game. I think it's fantastic, adding so much to the experience. But I can see how a lot of the fanbase considers Fallout (one) to be better at the end. The first game provided a tighter experience, with your character being on an important errand, leaving his/her underground home for the first time. It had this sense of shared exploration, where everything was new for you and your character. There was also this timer of sorts that kept ticking down to the annihilation of your people, if you wasted too much time running around they would die of thirst and that was it. The second half of the game removed the timer, but kept one focused goal that felt quite epic and daunting. When you finally faced the mastermind behind many of the storyline's bad deeds, you were going to be surprised. To me, the Master is still one of the bests villains I 've ever encountered in a video game. And you could beat him without shooting a single round, which made you feel that if you specialized your character in something that wasn't absolute slaughter, the game also had a spot for you. The game world was... I think "atmospheric" would be one word to describe it... you felt really immersed.

Fallout 2, on the other hand, was a bit more loose at the beginning. No deadlines and a massive map to explore at your leisure meant that you could and probably will get sidetracked from your quest. The locale was new, but many graphics and sprites returned from Fallout, and if you played the first you're not going to be surprised at the the depiction of life in the wastes. It felt familiar. Once and again you would see some cutscenes that would remind you of your quest, but it really didn't feel so pressing as the first time around. The game's pace sped up at the end, though, and for the better. Fallout 2's conclusion is really satisfying, including one hell of a boss battle that, sadly, couldn't be avoided like the one in Fallout.

I hope you find my opinion worthwhile. I apologize for the wall of text.