I haven't try all of those hopper yet, but I think this is the difference from the information I seen.
Short answer:
Normal hopper is the original hopper like we know, nothing different (I think)
Mob hopper only collects mob drops, and Crop hopper only collects crop drops, but it only need one in a chunk ( I think) to automatically collects all Mob/Crop drops that dropped within that chunk, regardless of its position.
Long answer:
I believe, the way Mob/crop hopper works is, just place one of the hopper anywhere within a chunk, then every Mob/crop drops that dropped in that chunk (killed/broken) will be automatically collected. The next thing is you can place Mob/Crop chests in that chunk anywhere and then the collected items will be "teleported" into the Mob/Crop chests, regardless of its position. Note that all of those must be in the same chunk. That's why Mob/Crop hoppers is more efficient than normal hoppers.
There are list of items that Mob/Crop hoppers can only collect. I think Mob hoppers don't collect Mobheads, but the other Mob drops are collected, while Crop hoppers collects all crops, cactus, sugar cane, and custom crops, but probably not Trees/Woods. I'm not sure if Crop Hopper collects seeds or not. The Mob/Crop hoppers can only be obtained from /shop (/ah probably can too). About normal hoppers, I'm not sure if it can only be bought from /shop or it can be crafted too.
Note: Don't forget the island's hopper limit (upgradable)
I'm not 100% sure about all of that, so if there was some mistakes, missed info, or additional tips and trick, please let me know. Thank you.