It is a common belief that in films the more sequels there are the worse the films become. But in gaming, games often improve with sequels. It make me curious. What are some instances were an original game was vastly superior to its sequels?

System Shock 1, imo.

Also' I guess you will get a few responses about how sequels for consoles were more shitty.

Ultima Underworld 1, also imo.

The more i thought about it, there actually are more original games that are better than the sequels. Classic case - COD series. Also AA was better than AC. DAO 1 better than DA 2. Many of the older series like Ultima's etc were better than there later incarnations .Current games -----> arguably older PC originals, of which now sequels made more to cater to consoles, the originals were better (graphics aside).

Imho Bioshock 1

Dragon Age:Origins

Bard's Tale 1. Dungeon Master. (not numbered but definitely superior to skullkeep)

Fallout - Isn't completely filled with 4th wall breaking jokes.
Crysis - Crysis 2 was substantially more restrictive and buggy
Lemmings - Not a huge difference, but the original tile sets were nicer; also, better difficulty curve.

I suppose this will probably be posted soon, so I guess I'll post it first: Deus Ex.

Personally I have no problems with Deus Ex: Invisible War, but most people likely hate it to the core.

There is a lot. Several mentioned already are great examples... Dragon Age, Deus Ex, Crysis, BioShock.

I will add Prince of Persia Sands of Time and, for Splinter Cell (1 is better than 2, 3 is better than 4 and 5).

Crysis
Fable

There was a noticeable change when PC series got sequels developed for the original XBox (Thief and Deus EX come to mind). Got a lot of scorn from die hard PC gamers.

In general I think it is very hard to make a good sequel to a great game. Either you create a unique gaming experience (UU, Fallout, Bioshock) and then the sequel is just "more of the same". Most noticeable in Fallout 2 or Ultima Underworld 2. But more good stuff doesn't make it better overall. Especially as you cannot recreate this feeling you had when you first played the game.

And then there is the "they changed it now it sucks" mentality when creators try to take a series on a new approach.

And interesting inversion is also Far Cry. 1 was better than 2 but three is better than both combined. Not really fair, as for all intents and purposes FC2 was a completely new game.

There is another scenario which leads to the second installment in a series being the best.

A developer makes a game with a new IP. They have a creative vision of what they want the game to be, but because of deadlines, delays, budgets, etc. they are forced to skip some of the things they wanted to do with the game. It may also be buggy and/or unbalanced. Nevertheless, the game does well enough to warrant a sequel.

This time around, they have a working base, so they don't have to start from scratch. This enables them to implement all the things they had originally intended, and so the second game ends up becoming what the first game should have been. This is of course a bigger success, so it's sequel time again.

Now however, they are outside the scope of what they wanted to do in the first place, but since you can't make a sequel identical to its predecessor, they have to come up with some changes, regardless of whether they fit the game or not. Often they don't, because they are "tacked onto" an already finished design, and so from the third game onwards, the series goes downhill, or takes a different direction than the first games. Taking a different direction altogether may be the way to go, since it may gain the series new fans to replace the ones who abandon it because it's "not what it used to be".

I find there are a lot of game series where my favorite installment is the second one. A few examples:
Civilization
Heroes of Might and Magic
The Settlers

This is very true. There are games were the original is somewhat redundant nowadays.

Is there really any benefit which Civ 1 has over Civ 2?

Mafia
Definitely better than Mafia II