It was an action/puzzle game for the Amiga and other platforms from Oxford Digital Enterprises, which became Empire Interactive Entertainment. Here's the Mobygames page.

http://www.mobygames.com/game/amazing-spider-man/screenshots

As you can see from the screenies, the game uses "single screen" rooms on a sort of grid map. The bar at the top next to the little heart fills up as you explore more rooms. Your health meter was picture of Spider-Man on the side of the screen that slowly turned into a skeleton from the bottom of his feet on up.

The reason I'm asking about this game is because I swear I saw at least one or two other games that used the same "engine", and I'm pretty sure it wasn't Impossible Mission. Does anyone know what they were?

Can't say I know anything about this game, but it looks like it could be pretty good. I have no answer for your second question, but I wish I did. :/

Whoa, I DID play this one back then on my PC. I remember it being real fun in a retro-spectrumish kind of way. Spidey was able to stick to walls and ceilings, and also shoot his web in all directions and paralyze enemies. With a little skill you could even have him swing from the ceiling in some screens.
Hey GOG, you think you could get hold of this one? I don't think it'd be a blockbuster but I guess must be an easy one in regards to distribution rights and stuff.

It'd be harder than you think. Marvel and all it's characters are now owned by Disney, so GOG would have to also work with them to get the rights.

Didn't the old Amiga Batman game use the same engine?

Oh, that little thing. Sure, i've played it! On the PC and on the C64. I could never finish the game though, due to it's length and quite confusing map design.
Never knew there were similar games, but that's cool, i liked the look of this game, with ot's tiny tiles. ;)

Yeah, I played it, I loved it, I hated it. Freaking weird game, indeed.

I'm still trying to purchase a boxed copy of it, with no luck thus far.

If you're talking about the game based on the 1989 film, then no, it wasn't. And besides, that game had only two platforming levels, two chase-view vehicle levels, and one (maybe two?) timed puzzle level.

If you want to get ahold of the game your best bet would be to buy a copy of Amiga Forever, learn how to use WinUAE, or whatever Amiga Emulator works on your OS, and find a site that serves up ADFs. I'm sure you could find a torrent of the complete, 22-odd GB TOSEC collection somewhere.