Disney is NOT cheaper than carnival, but it is very much better. Carnival is a party company--lots of smoking, drinking, casinos, etc. They are trying to change their image, but they are the classic 18-25 year olds who like to party hard and not always legally.
Disney--MUCH cleaner, no casino. Very little smoking, little or no offensive drinking. Great family atmosphere. The adult areas are clean, safe, comfortable, etc. I've never felt uncomfortable on a Disney ship. Can't say that for the other lines we've sailed.