Both Carnival and Royal Caribbean offer a similar product. You would both probably enjoy either.
You would have to choose for yourself, by going on both (great excuse for a second cruise)but I would suggest you let the choice be guided by the cost you are willing to pay (Carnival tends to be a little less expensive) and by the itinerary.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise
NCL Dream October 2006, RCCL Mariner OTS June 2007