Examples of this genre include Cave Story, La-Mulana, and Aban Hawkins & The 1000/1001 Spikes, all published (or almost published) by cave game fans at Nicalis. Other examples include the two games I want to talk about, which share a lot of DNA but diverge in interesting ways: Mossmouth's brilliant Spelunky, and one of the progenitors of the genre, Spelunker (and its modern-ish sequel Spelunker HD). Spelunker was first released way back in 1983 as an Atari computer game; it later found success and infamy in arcades and on the NES.
Mossmouth's XBLA game has been a critical success since its release in July, and was beloved as a freeware PC game before that. Conversely, most people don't have anything nice to say about Spelunker, and I think my appreciation of the PS3 sequel puts me in a fairly elite club. However, that club most assuredly includes Spelunky creator Derek Yu.
Aside from the specific elements Spelunky took as inspiration, it also captures the feeling of playing Spelunker: of having to move slowly, deliberately to avoid the myriad terrors that could kill you at any moment. Both games share the thrill of just barely scraping by, surviving hostile environments and impossible circumstances.
The differences go beyond variations on a theme, however. Spelunker/HD is actually much less forgiving. You don't have four hit points, just one, though you have multiple lives. Furthermore, there are more things that can kill you than in Spelunky, and for seemingly less severe mistakes. In Spelunker, your air supply constantly counts down via a meter on the screen, and you asphyxiate unless you regularly pick up air tanks. That air tank also powers the air gun you carry, the one weapon you have against ghosts. You have to fire it quickly, by the way, or the ghost will still kill you in the middle of its "disintegration."
What I suspect turns some players off of Spelunker is fall damage. Whereas in Spelunky you can fall 50 feet or so and lose one HP, falling any distance further than the Spelunker's body results in instant death. That means you could drop from a platform you could easily handle in real life and die. You could jump from one rope to another, and not catch on early enough, and die. He is comically fragile.
Despite the similar elements, the games have different goals. Each level in Spelunky is randomly generated, meaning you cannot prepare for the specific flow of a level. You can't know what's coming. In Spelunker, on the other hand, all of the levels (and there are 100 floors in Spelunker HD to start) are designed, and all of the encounters scripted. It is possible, and necessary, to get better through practice. Where Spelunky is a more sophisticated game, all about experiencing a unique story caused by unexpected interactions between randomly juxtaposed moving parts – "I threw a clay pot at a snake, but it hit the shopkeeper instead, and he freaked out and jumped at me, then accidentally impaled himself on some spikes, and now I have his shotgun" – Spelunker is a more straightforward platforming experience about just surviving against crazy odds.
They're very different, but I think both are worth experiencing, especially if you've played one and enjoyed it. It's not often you get to see the same idea explored in such divergent ways. And if you do want to try Spelunker HD, now is the best time. It's on sale through PlayStation Plus, and just got a bunch of new levels as DLC, which are also on sale. So at least the purchasing part isn't impossibly painful.