The list could be very big for this, even if the original isn't necessarily "great", ground breaking, or revolutionary within the industry (and for its time). There's still so many sequels that are better than the original (or the "previous" title, not necessarily being the first, for example in a trilogy when the third is better than the second, while the second is also better than the first). I could easily start from the 8-bit era all the way to the PC platform with nowadays games, and even passing by Arcade exclusives (even if I try to keep the list for "PC" games it'd still end up quite a large list).
I'll name a few though for the heck of it, and in no particular order. The following is quite subjective as you can imagine. Just a side note though, just to show how subjective this will be, Craig234 mentions the following: " For example, I'm not listing Fallout or Portal, because the originals were great enough that while I'm glad they made (better?) sequels, they aren't that much better to be on the list. ".
Well I for one think that both Portal 2 and Fallout 2 are significantly (much) better than the originals (in fact I got bored of the original Portal very soon and "forced" myself to complete it, when for the second I couldn't stop playing until the end), and I would gladly put them on the list (well pretty much just did by saying this). So of course it's very subjective.
Most of the games I'm listing are older games, since I know that most people will concentrate on much more recent (and probably mostly PC or ported to PC) games, so here goes...
º Super Mario Bros. 3
I don't think that any such (or similar) lists should go without it. It is one of the various quintessential exemplary games showing what "a sequel" has the potential to be, or even "should" be, in comparison to the original or its previous installment. Not necessarily in terms of "story" aspects but also (and can be isolated to) mere game-play mechanics and features addition and/or improvements.
Yes, of course the original set new industry standards, and probably single-handedly rejuvenated the whole video gaming market, but in terms of game-play mechanics evolution in comparison to the rather unoriginal second title (namely the "it was a dream!" sequel) SMB3 "did everything right, and some more". It was without a doubt not only the best of the then-trilogy of SMB games but was also the best on the NES and any competing consoles for the following year+, arguably only dethroned by Super Mario World two years later.
º Mega Man 2
I do not think that it is even arguable that had it not been for that very sequel alone the whole Mega Man franchise would have never been anywhere close to a sequel at all. In other words... the original - while appreciated enough - was nothing so spectacular even at the time. But when MM2 was released it sent a shockwave that even Mario felt pretty good. It was a great "competitor" to other platforming games and franchises and Capcom understood then that they had a jewel in their hands.
There is a lot of nice story bits about MM2 but its Wikipedia page does a pretty nice job detailing the most important parts of what made it the great sequel that it was then and still is today (when you look at it in retrospect if anything, or you can still respect its superiority in every single way to the original regardless of the titles that followed it).
º Mortal Kombat II
Again, known history for the original (very violent, broke taboo of what was acceptable in a video game, responsible in part for the creation of ESRB and its finishing moves a dream come true for anti-violence and anti-video games activists). Also, of course, known for its overall amazing game-play at the time and the first true competitor to the Street Fighter franchise.
But Mortal Kombat II - while inevitable as a sequel per se - is where the "true" MK game-play began, that is in my opinion. Yes of course the original "was the true beginning", on paper. But I'm pretty sure that Nintendo were happy to see sales of the sequel on their SNES compared to how the original ended up being a gold mine on the Genesis for obvious reasons. And, ESRB? Who cares, let's make MKII even more violent, oh and of course more fun, too!
It has more, better moves, the combat pace and recovery time after each moves was faster, even allowing "unofficial combos" to be made (improvised first, then later became known combos). It had more, better levels, more graphical details, better colors palette (even on the Genesis and SNES versions), more voice acting, better "story", is generally grittier of a game (excluding Babalities) and is - overall- simply more... let's say... toasty then the original. It was the accumulation of the original's success which gave birth to one of the "greatest sequels of all time" (in my opinion).
º Sonic the Hedgehog 2
Simply put, it was to its original what Super Mario Bros. 3 was to the first. Now yes I know many would argue that Sonic the Hedgehog 3 was better. Well yes it was great, I would probably say just "as good as" the second. But in comparison to the original (even if the thread does not restrict comparisons to the very first in a trilogy for example) then I believe that, definitely, STH2 was a significant leap (for the best) and that, overall, STH3 was not that much of a leap when compared to the second and itself.
º Turok 2: Seeds of Evil
Cerebral Bore, enough said?
Nah ok I'm kidding (well the Cerebral Bore was disturbingly gory and fun to use, and watch in action). But yeah, the original, Dinosaur Hunter, was very popular. It sold very well and was a general critical success. I agree with all that, and to this day still love Dinosaur Hunter (and I got my PC version installed right now).
But despite the undeniable fun that the original was (and still is today) I still consider that its sequel, Seeds of Evil, is amongst one of the very best sequel of all time. And, no, the checkpoint saves system did not get on my nerves and no the frames-rate slowdowns were far, far from being anywhere near a problem or even a "distraction" for me generally speaking, with the only few exceptions to that in parts of the Hive of the Mantids (fifth level).
I think that Turok 2 did everything better, added upon it, was even grittier, more violent, had more and better weapons, superb animations (that still hold their ground to this day) and one of the best soundtracks ever made in any FPS games I can remember (if not one of the best soundtracks on the N64, period).
º Resident Evil 2
At least at the time, it was one of the best exploration, puzzle-solving and survival horror games I had played (not that there were many such games at the time to start with). The original was fun but I have to admit I played the first RE a couple of months after I had completed the sequel.
So my first "experience" with a survivor horror game was RE2 and probably developed bias for it due to that. But I have completed the original two times (and the sequel probably a dozen times, and on all four scenarios with both characters) and I'm convinced that RE2 simply did everything better, especially the overall atmosphere (I.E. level design, game's setting, enemy placement to scare you, or passing by creatures on the outdoor side of a window from a freshly-loaded corridor, etc... it was all hand-crafted level of quality areas). I must say though, I never got to buy it at the time but I later played the N64 version and that one just cemented my thoughts even more on how brilliant the game was (and the N64 version being even better than the PS1's).
º DOOM II: Hell on Earth
Well, sure Ultimate DOOM was awesome, but I don't think I really need to describe why or "how" was DOOM II "much better" on pretty much every aspects.
º Grand Theft Auto III
It was an important enough sequel on its own that many gamers were not even aware of the GTA franchise at all before GTAIII was released. I won't write a novel about it but I think that its success and what we now know of (of what the franchise is about) the GTA franchise today explains itself. Now I admit I got bored real fast of GTAIV and probably won't get anywhere near the fifth one but at the time and along with San Andreas (and Vice City as well but to a lesser extent for me) it was quite the bomb in the video gaming world.
So why GTAIII specifically? Well because as mentioned (and I was part of those gamers) prior to it I had no idea whatsoever about what GTA was at all or that two previous games of it even existed. But I did get to play them both much later and perhaps that was the problem. They did not have much "impact" at all on me and wondered even why they bothered with GTAIII in the first place since I thought that GTA and GTA2 had never been really popular.
Now while I think that (when thinking about it now) GTA2 was quite better than the original on its own was still clearly pushed aside by GTAIII almost as if both previous games indeed never existed anyway. It was as though GTAIII came out and the devs said "ok finally, technology allows it, THIS is what we wanted from the start, forget the other two please" (that's my impression anyway). And am I "glad" that they made GTAIII? Hell yeah! I loved that game at the time, but today... not my stuff anymore, but sure looking at it in retrospect I can't exactly deny the dozens (literally) of sleepless nights I was "forcing" myself through to play this game especially on week-ends with my cousin.
º Half-Life 2
I tried not to mention it since obviously the original represents so much for the PC gaming industry that it pretty much became (and still is) an icon on its own. Was Half-Life 2 even better than the original? Well, yes. It was (well that is, of course and I like to repeat that one, in my opinion). I absolutely adore (and praise) the original, but I think that it was more of a technical revolution - overall - than a true leap in "mere" game-play features and mechanics. That part, I believe, truly belongs to Half-Life 2.
There's enough history on Half-Life 2's contributions for the gaming industry as well (along with its Source engine on its own, regardless of Half-Life 2 itself). But the game-play of the sequel, compared to the original, is light-years ahead, a quantum leap (especially when you consider Episode One, and mostly Two, in the bigger picture).
The animations, the music, the masterfully-scripted scenes (without CGIs) to tell the unfolding story dynamically (again, without damn cut-scene "interruptions" like it is the case nowadays in almost any games that tries to be a movie instead). The facial animations portraying emotions like no other games ever came even close to manage (lest being in a CGI movie) along with the physics used as an actual game-play mechanic, not to mention the very quantum leap in general game-play mechanics evolutions on its own that it represented in the industry...
I don't think that I can keep giving that game more hugs and praise even after all that time, Half-Life 2 is one of those "timeless" contributing (to the gaming industry), distinguished and evolutionary (and revolutionary) FPS (and PC) games that we'll be talking about for the next decade (and beyond). It was also simply put a much better game, as a sequel, to its original counterpart, even if the first was as important during its time as the sequel was in its own period as well. They were both great games, but to this day Half-Life 2 stands proud on the podium of FPS gaming, PC games in general, and just being one heck of a great sequel.
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If I keep participating in threads like this like I've done recently then I think I'll end up on a serious buying spree for older games and platforms (consoles), and my wallet will come haunting me in my nightmares for doing it (and I suspect it'll have my stepmother's voice to make matters worse, also I don't have a stepmother, but the wallet will pretend I do just to piss me off).
And I need to go sleep now (good thread by the way).