The Addams Family and Addams Family Values.
The Road Warrior and Mad Max:Beyond Thunderdome.
Sequels to Mad Max. Both of them better than the original. The original had a limited budget and a god awful dubbing in the U.S. release.
The sequels have a better budget, and show how the director had improved in the following years.
Actually, I liked the first Alien much better, but it’s hard to make a solid case either way, as they were so different. Alien was a horror/scifi movie, Aliens was an action/scifi. Same thing with the _Evil Dead_s. Switching generes probably makes it easier to avoid feeling like a copy . . .
Hmm. Empire Strikes Back and Samurai 2 are easy. Maybe The Swordsman, though I saw Swordsman 2 back to back with The Magic Crane very late at night, and they might be blending together to form one meta-kungfu movie . . . Once Upon A Time in China 3 (Though Number Six beat me to it) was much better than the second one, and might have topped the first, though I’d have to see them both together to judge.
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“This is really disapointing as a followup to Psycho . . .”
Neverending Story 2 but not Neverending Story 3 (yuck!) or 4 (double yuck!!)
I don’t know about Mad Max Beyond Thunderdome. That movie is terrible.
Ura-Maru: I can’t beleive I missed Swordsman 2 on my list. Good catch! It’s defenitely better than Swordsman. It would be difficult not to improve on the first when replacing the leads with Jet Li and Brigitte Lin.
Oh, and I’d like to add Sanjuro to my list. It may not be better than Yojimbo, but it certainly is as good, and easily qualifies as one of the greatest sequels ever.
Also, Manon of the Spring, the sequel to Jean de Florette. I know, they’re really two parts of a single movie, but so are the first two Godfather movies. Blending in the following story with the previous one is what a great sequel is supposed to do.
Once you buy into the joke, The Spy Who Shagged Me is just as good as Austin Powers: International Man of Mystery, IMHO.
Deathstalker 2 was much better than the original Deathstalker!
Where are you all going? Was it something I said? 
They made a Neverending Story 3 and 4? Ewwwwww… but at least they’ve lived up to the movie title…
La Dolce Vita originally started as a sequel to Federico Fellini’s I Vitelloni (working title: *Moraldo in the City); the similarities and differences between the two films is enthralling.
Terminator 2 by far… plus (although in my opinion a “girly” movie) The Jewel of the Nile.
Oh come on. The original was so much better then #2. It was so much deeper. Number 2 feels like it was just produced for a quick buck, IMO.
As for sequels as good as (or better then) the original, I can’t think of anything not mentioned. I know it’s a little off topic, but as an avid cartoon fan, I have to say “The Land Before Time” #'s 2-7 were together perhaps the worst sequels ever when compared to the original.