I usually describe MUDs and MUSHes as telnet accessible text-based rpgs.
And then people tell me, "What did you just say?"
What that means, is it's a text-based roleplaying game, that you can play with any telnet client.
A telnet client would be something such as zMUD or Pueblo. Now, the difference between a MUD and a MUSH is fairly simple.
On a MUD, you kill mobs for experience; a mob being a monster(mobile). MUDs are mostly killing, with very little roleplaying involved.
Whereas on a MUSH, the entire thing is role-playing oriented. I've only played on a few MUSHes, and none of them had any mobs at all.
I myself prefer MUDs, not because I don't like roleplaying, but rather because I've never seen a MUSH with enough people to actually roleplay.
MUSHes get boring fast without people, and being bored is not my style. So, to sum that up, MUDs are faster and more action packed.
And my favorite MUD is Aargh! So, which you play just depends on what kind of game you're looking for. If you want some hack'n'slash, with conversation, go for a MUD.
If you want pretty much all conversation and roleplaying, MUSH is the way to go.
Or at least that's the way I understand it. ;)