I have cruised both but I prefer Royal. Their Freedom class of ships have so much to do and there truly is something for everyone. Even on a bigger ship the Royal staff made us feel just like royalty. They remembered our names, where we were from and what we prefer to drink. I didn't feel that with Carnival.
It depends on your preference. Carnival has more family-oriented cruises, while RC is more upscale. Each has its own quirks. My own research always led me to RC most often because they seemed more adult-oriented. I recommend browsing cruising message boards to learn members' experiences on specific cruises.
We have cruised on both lines and we prefer Royal Caribbean. The Carnival cruise had too much drinking and partying for out taste. One man was seriously injured when another passenger pushed him down the stairs. The cruise itinerary had to be changed so he could get to a hospital.