I think that Royal Caribbean has cleaner ships, better ports, nice interiors (but every ship is the same) and has some extras that other lines dont have (Ben and Jerrys, rockwalls, Johnny Rockets...) But , their food is not nearly as good as Carnivals food, and they just seem to lack the "party" atmosphere.
Carnival has a exclusive fleet of ships, with everyone having a different theme, which also sets a party mood. Their nightlcubs are alot better, and the food is 100x better then RCCL.
They are very simuliar as they target basically the same demographics. My personal feelings is that Carnival has better food and entertainment and the standard cabins are larger. We also enjoy the service better as the staff is more friendly and less formal. Understand these are GENERAL statements and not true of all the ships and all the time. RCI has prettier ships, more onboard activities such as rock walls, golf, pooltables, golf, basketball/ In addition the staff is more polished and service is on a bit higher level. Again, these are all GENERAL statements and not always the case. I also find it more difficult to get around on a RCI ship as they tend to try and funnel you through the casino and that is not only a tight squeeze when there are hundreds headed in one direction, but I also do not like the smell of smoke.