Hi - Disney was our all out best cruise. We have not done Royal Caribbean but have friends with children who have and they loved it. So I think you cannot go wrong with either one.
I highly recommend Royal Caribbean's Freedom-Class ships that feature the Dreamworks Experience. Just as good, if not better, than Disney.
Disney is wonderful, just very expensive... I'd go with RCL, they have awesome things like flowrider ( their surfing simulator), great kids programs... and you're not paying the Disney premium.
It really depends on how old your kids are. If the kids are young, Disney is always a solid choice (if you don't mind the high prices), but once they get a little older, I think RCL might be a better bet.