Hello. I've sailed on both and we've enjoyed every cruise we've taken.
On Disney, there are a few perks that we enjoyed. The pools are not salt water. They have a movie theater that runs new films. The interior cabins are slightly larger, with the bathtub that Cruise Fanatic mentioned. The shower is an issue for some young kids. Excellent service. The shows are spectacular. My five-year-old loved the shows and meeting the characters made her whole trip. My family does not drink enough soda to make the soda card worthwhile, or to make Disney's free soda a big deal. The dining room staff on both lines make a special effort with the kids, at least in our experience.
I can't compare the kid's programs, cause my kids usually won't go to them. Disney is overall more family-oriented. There is an "adults only" section on the pool deck. There are bars, thank goodness, but I did not see the drunken free-for-all that usually entertains me on Carnival's Lido deck.
For us, Carnival is our regular vacation and Disney was probably a one-of. We might do it again when we can afford the new Northern European itinerary.