As I recall, differences between the editions are more fiddly than anything else, mostly corrections and revisions. Until you get to the Mongoose (?) edition, which also shows you how to run it as a straight grim dystopian game.
Which is nice and all, but kinda misses the point. (And anyway, all the best jokes are for the GM.)
Wikipedia has a nice summary of the various editions.
It is worth noting that the 2004 XP edition, which introduced the possibility of running it straight, was done with full approval of the original designers who viewed the later West End editions to be too cartoony in their presentation of the satire. The first edition was a much darker, less humorous take on the world than second edition. I would guess though that for most of us who like Paranoia, the feel of second edition is what we tend to remember. 2nd edition was certainly the system I played back in the day.
The latest edition by Mongoose is a bigger departure from the original rules than most of the preceeding editions in that it introduces card combat and a d6 dice pool mechanic.