We have done both RCCL and Princess. I'll try to compare for you:
RCCL: Voyager of the Seas and Radiance of the Seas.
Pros: Beautiful ships, HUGE, lots to do (almost too much), more families, better itineraries, kids programs were awesome
Cons: ok food (except in Johnnie Rockets: great food), service was friendly, but did not go "the extra mile" that Disney does, rooms are not as big as Princess. They have extra charges for nearly everything.
Princess: Grand Princess.
Pros: Well thought out floor plan, never seems crowded, good food and nice selection, seems to be an older crowd, Service was impeccable, rooms (we had a minisuite) were very comfortable, LOTS to do.
Cons: didn't like the itin as much (eastern carib), older ship; shows some wear and tear and creaks alot, kids programs not as much fun as Disney and RCCL.
See if that helps a bit. I have also heard good things about Celebrity. We have also traveled on Carnival and did not like the experience...but that was five years ago and I hear the "party atmosphere" has been toned down. Of course that is not what my friend said who just sailed on the Destiny!
Any more questions, I'll be glad to help if I can.
Lisa