I don't have kids, but I can tell you from the kids that were on the ship I was on in January (Carnival Victory) were having a good time.
If I had to choose between RCI or Carnival, I would go with Carnival hands down. The price has nothing to do with it, but what you can do makes it. I do not ice skate, I do not surf, and I do not like to climb rock walls, which means that most of RCI's ships wouldn't appeal to me. Sorry folks, if I am on holiday I don't want to goto a cooking demonstration, but rather would go watch the silly games/activities on the Lido deck on any Carnival ship.
Carnival cabins are also a generous size for the price you pay. I told my mother, after she asked me why I chose Carnival and not RCI (we had Legend of the Sea docked with us in Costa Maya) that the size of cabin we have on this ship would be at least $300-400 more per person on the RCI ship. I also said I know how much fun Carnival is and I knew I would be hard pressed to get tanning time in between all of the activities that go on a Carnival ship.