I don't have Doomsday Eddie, but I do have a few troopers who let me use him so I'm somewhat familiar with his abilities. From my limited experience using him, I have to admit he's probably the stronger of the 2, but I'm a huge fan of Rainmaker Eddie as he was my main guy for arena fights until I got Hallowed and Breeg Eddie, but I still use him for many battles depending on the team I'm attacking. As such I'd like to point out the benefits I've seen in Rainmaker, aside from the obvious high damage.

•He has extremely high ATK/MAG stats, which means if you don't have Eclipse talismans you can use 6 Resilience talismans to give him a reasonable amount of health without having to sacrifice damage <-- this might be his greatest advantage

•His fury move deals devastating damage, especially if there are a few marks on the target, plus puts shroud on your whole team
•He has an inexplicably high Special stat and gain extra turns, which helps him to build up to that (potentially game-changing) fury move quickly
•His 6 power-cost move Stab can remove 4 effects including golden ones while applying permadeath, which is great to use against teams with a particularly dangerous damage/debuff dealer and one or more reviver (plus it gives him an extra turn, as Edsel pointed out)
•If you have another character that can stack a lot of marks, Rainmaker can double those marks with his Rain move (which I believe costs 5 power)
•He has a basic attack that is guaranteed to give him perfect immunity, which is nice for dealing with primordial teams, or douchebag stunners like Horus
•Another basic attack has a decent chance at turning golden effects into marks, further improving his ability to counter primordials or eternals

•His main (possibly only) weakness is that he is pretty easy to kill unless you have eclipse talismans or are able to fill your fury bar before he gets 1-shotted. Full resilience talismans or immortality help to mitigate this, but its best to have a strong reviver to back him up. I usually run him with The Clairvoyant carrying destiny talismans to keep him up, and The Alchemist carrying avenger talismans to make it difficult for enemies to apply permadeath