Disney will win out for food and service, especially for kids who are picky eaters.
But when it comes to onboard activities, you can bet most children over the age of 8 will actually have more fun on Royal Caribbean Freedom than a Disney ship.
Disney cruises carry the Disney environment over to their ships, but obviously without the rides. They have Disney characters everywhere and great entertainment.
However, RC Freedom has more activities older kids will enjoy such as rollerblading, rock climbing, a water park, huge video game parlor, mini-golf, ice skating (!) etc. Royal Caribbean is a hard act to beat, especially Freedom which is twice the size of the Disney ships.
A lot depends on the age of your children - younger kids = Disney. older kids= Royal Caribbean.