I don't have kids, but I have sailed on Carnival, and one MAJOR difference is the Adult Only areas that are on
DCL...Adult only pools, hot tubs, Adult only restaurant, Adult only beach. I am sure all the kiddies were having fun, but on DCL I never saw them! And Quiet Cove/Serenity Bay, and especially Palo are to DIE for!
I also liked the family atmosphere (sometimes things on Carnival got a little rowdy/raunchy..shows/partying, etc).. granted nothing to bad on Carnival, but I am a G and PG type of gal, and like all that DCL does to keep things that way.
I also liked the highly-themed rotation dining on the DCL, and thought the ship (I was on the Wonder) as a whole was a lot nicer then the Carnival Cruise. I am a Disney-holic though, so that might make me biased to the hidden mickey's and Disney Decor.
State Rooms are also 100 times nicer on DCL. Carnival, it was like a closet. Not that way on DCL, I had a Cat. 8 last time, and it was very spacious... felt like I was in a hotel room really, not a cruise stateroom.
I guess if I had to narrow it down to two main reasons why I sail DCL rather then Carnival--Adult only areas and state-room accomidations...Hope that helps.