The Carnival Breeze (March) and the Disney Fantasy (May) were my two most recent cruises! I rated each of them using the same scale if you are interested in reading.
Carnival Breeze -
http://www.victoriaonvacation.com/2014/04/2014-carnival-breeze-southern-caribbean_29.html
Disney Fantasy -
http://www.victoriaonvacation.com/2014/06/DCL-fantasy-rating.html
It had been a few years since I had last sailed Carnival, and I returned to them only because I was one of the lucky ones to receive the free cabin offer. That being said, I was glad to find that Carnival had really made some improvements since I had last sailed the Dream in 2010. The Breeze starts to incorporate the Funship 2.0 enhancements which aren't yet available on the Dream. I really enjoyed the Red Frog Pub - such a great Caribbean atmosphere with fun drink specials like the beer tube and fishbowls. Guys Burgers and Blue Iguana Cantina are two new complimentary dining venues on the pool deck. Bonsai Sushi is an additional (nominal) cost, but we enjoyed eating there. The kids seemed to really enjoy Hasbro the Game Show, and I found it entertaining enough to watch (mainly because kids do the darndest things). As for features also on the Carnival Dream, I love the cantilevered hot tubs on the promenade deck, and WaterWorks is great fun.
As for Carnival Breeze verses Disney Fantasy comparison...
Design
DCL Fantasy without a doubt wins in overall design. I ranted a bit in my rating about how the design on the Carnival Breeze is all over the place and does not transition well from one area to the next. The Disney ships overall are just designed with much greater care than any other cruise ship I have been on, but the leap of quality is especially obvious when it's sitting next to the Carnival Breeze.
Dining
Carnival Breeze actually does well with dining. I wasn't a huge fan of the atmosphere in the main dining room, but the food was good. I do not know if they have switched their menu yet to the American Table menu that they are working on incorporating fleet-wide. Atmosphere on the Disney ships is more interesting with the rotating restaurants, but I don't think Disney's food is anything to write home about, either. Breeze has a competitive edge when it comes to the casual eateries... Guy's Burgers, Blue Iguana Cantina, Bonsai Sushi, Fat Jimmy's BBQ. DCL Fantasy only has Flo's V8 Cafe which doesn't have nearly as much variety.
Entertainment
Disney Fantasy is clear winner here, especially if you have children. Disney's theater shows are of high quality and greater caliber than I have seen on any other cruise line. Carnival's theater productions actually took a hit after the Funship 2.0 Upgrades, replacing live sets with LED backgrounds and getting rid of the Carnival dancers, relying on a team of six singers/dancers for the entire production.
Adult-Only Areas
I can't rate kids' things because I have no children, but I can rate the adult-only areas for you. Carnival and Disney both have complimentary adult-only pool decks (some other lines charge) - Serenity for Carnival and Quiet Cove/Satellite Falls for Disney. Serenity is only a pool deck with two hot tubs. There is no adult pool. The chairs are very cushy and they offer a variety of shaded and non-shaded loungers. It's a rather small area, though, and finding chairs can be competitive. If you don't go out early in the morning to stake your spot, don't plan to find one until late in the afternoon. As for Disney's Quiet Cove and Satellite Falls, even during peak sunning hour, there were always available loungers. Some are cushioned and some are not, but I never had an issue finding an available padded lounge chair. Also, there is the Quiet Cove pool and Satellite Falls pool. The entire area is about twice the size of Serenity on the Breeze.
As for inside the ship, Disney's Europa is an entire section reserved entirely for adults with five bars/lounges. Carnival Breeze has the Punchliner Comedy Club (difficult to find a seat in) and Liquid Nightclub.