As people are asking.
Terris - This was developed by a player called Vanderghast who played the Sheffield version of Avalon. He told me once that one time he complained to Genesis about a particular issue with Avalon and the responce was "if you don't like it, design your own mud". Terris was the result.
Achaea/IRE - Designed by Matt Mihaly/Lazarus - Originally, relatively briefly licensing hourglass, then an engine called Vortex designed by Ben Maizels/Nostradamus. This was in use while their own Rapture engine was developed. Shortly before Rapture went 'live' Matt bought out Vortex and essentially closed down Cardea and Orone, at least for a while. Lazarus was still minorly active in Avalon for a while once Achaea was in beta before dissapearing completely.
Cardea - Macros and Arikarr are/were the principal developers of this game, Alecto certainly was another big contributor but not at the outset. Wraith is/was the same Wraith from Avalon. Original players drawn from Avalon. This was another game running on a license of Vortex while Macros developed his own engine. Was forced to switch to their own engine before it was quite finished due to the sudden unavailability of Vortex.
Orone - Designed by Archon and Kayhlan from Avalon, but with heavy contributions from it's original players who were drawn from Avalon and Achaea. Cleindori sticks out as a big contributor, but there were lots of others. A number of incarnations in a somewhat up and down existence. Originally using a license of Vortex, it was closed down when Matt Mihaly purchased Vortex outright from Ben, the game was then redesigned in an engine of it's own designed by Archon. Closed down permanently a few years back.
Evarayn - Don't know full details on this one, but apparently it has some connection to Orone. The skills are all from there and unless things have changed it's using all the Orone gameworld too.
Akanbar - As stated is run by ex-First Age players. Don't know a whole lot about this one.
Elysium - If I remember correctly is an original engine designed by British ex-Avalon players.
Some of the more striking differences between these games and others are the multi-layered skill and combat system, the combat system based on the ability of the player rather than the size or skills of the character, the complex player-run organisations, gods as players and not simply administrators, the learning system, etc. I could go on.
As pointed out by Narissa, Avalon still uses hourglass and it's a notoriously unfriendly engine. A couple of errors though, Apollo has never been a coder and Magellan certainly was a long time player, in fact he was still a very active guildmaster and baron of a certain city for most of his existence. I don't understand the Alessandria reference? Most of the coders for Avalon have been players, alas this has been the main problem with most, they spend their time coding things for their friends or themselves.
Also, as people are interested in the history and development of Avalon and muds, these links may be interesting.
http://www.mud.co.uk/richard/imucg4.htm#s7 - An old review of Avalon, interesting for historical reasons.
http://www.mud.co.uk/richard/rpifeb93.htm - Another interesting page, I seem to recall reading Anarchos took over Avalon at some point.
I'm fairly confident with what I've wrote but if anyone would like to correct me on anything, feel free.