Jaguar Doom: The basis for nearly every other port, identifiable by simplified geometry and texture variety (the telltale trait starting out is the second room on your left in E1M1. In the PC original, there's pillars here and a variety of textures; on most ports, it's a simpler, flatter room with a grayish look, and no pillars.) John Carmack headed this port up. Famously lacks any music at all, but it runs quite smoothly. Some missing or altered levels (this is standard) and a few levels of the original have been replaced with new ones.
Sega 32X Doom: A very divisive version that actually came out before Jag Doom despite being based on it. It features a far lower resolution, and control and gameplay simplifications. However, it actually runs very smoothly, one of the best ports of the 90s in that regard. So why is it divisive? The music is notoriously bad (some say the developers somehow messed up or weren't able to utilize the full sound capabilities of the system), and it's missing nearly half the game, with the entire third act missing. Instead it just abruptly cuts to credits after completing a not very climactic second act level, and if you used a cheat code... you get a C:\DOOM MS-DOS prompt text message.
SNES Doom: One of the oddest ports; it actually has more accurate geometry than the Jaguar-derived ports, but it has no textures on the ceilings or floors. It has many levels but things like shotguns being pellets, strafing, and monsters turning around (instead they just always face you) are gone. It runs at a super low resolution and molasses framerate, even for a SNES game. On the other hand, it's a technical marvel that it runs on the system and the soundtrack is a great rendition. Oh, and the programmer went on to create Bleem!
PlayStation Doom: Considered almost certainly the best home version of the time overall, but a weird one in some respects. It uses colored lighting and a completely different soundtrack and set of sound effects to make the whole game more gritty and ambient and atmospheric. It includes a bunch of levels from Doom II. If you bump up the difficulty, then Doom II exclusive enemies appear in some Doom I stages. It also has edited, changed, or replaced (with console exclusive) stages. That said, it has a staggering amount of content (much of Doom 1 and 2), runs fairly smoothly, and even supports system link for LAN play.
3DO Doom: There's an incredible
interview and
article about this. The programmer for this project, Rebecca Heineman (an industry veteran who IIRC worked on Wolfenstein 3D for 3DO) was tapped by a company that had gotten the Doom for 3DO license and was expecting a quick windfall. She soon learned that despite promises that the game was up and running, they had absolutely nothing, forcing her to ask John Carmack for the source code. The publisher made incredible promises about new weapons, levels, enemies, and FMV sequences (photos of these scrapped live action scenes recently emerged and they are INCREDIBLE). In short: they were the kind of guys who thought you just drew a level or gun and put it in the game, easy peasy, like an hour's work, right? At one point they even asked her to just download some fan levels from the internet and slap them on the disk to fill it out. The game runs like garbage in a tiny window, with options to enlarge or shrink it to balance the framerate (there's a secret code to make it fullscreen, which was put in in the hope's that the 3DO's successor, the unreleased M2, could use it). It's essentially the Jaguar version, but running like garbage in a tiny window. It has music though! She found a garage band that did a really kickass version of the soundtrack! That's about the only good thing about it!
Sega Saturn Doom: Imagine the PS1 version. Now imagine it has all the colored lighting gone. Now imagine that it has the most inconsistent framerate in the world. That's this game. The programmer has gone on record that they had a version running that used the Saturn's unique hardware to do hardware acceleration, but the visual effect it produces (textures warped due to affine texture filtering) made John Carmack give it the thumbs-down, forcing them to essentially run the game in software mode for a massive performance loss. It's also quite buggy and unoptimized, leading to a framerate with massive swings.
There's also the GBA version, the version included in Doom 3 on Xbox, and the XBLA version for 360, but I need to stop for now.
Oh, and Doom 64, but that's an entirely new game made by the PSX Doom people.
edit: also, these are only the general differences. there's a LOT of differences when it comes to level layout, which levels were maintained, which were altered or were completely new, which enemies were included and which were cut, enemy and weapon behavior, etc.