Since you are cruising with your family/kids, I'd suggest any of the Royal Caribbean "Voyager" class ships. They have more to offer kids and just about everyone, while the Grand Class are also nice, the Voyager class are geared more towards kids and people of all age groups, the adults will also find more activities.
these two cruise lines are very close in comparison. I would give Princess higher marks for food, and service. Both have very good kids/teen programs. The Voyager Class ships have more "bells and whistles" shall we say. If you want the rock climbing walls, the inline skating tract, ice skating rink and a main street parade with a Mardi Gras type atsmosphere then the Voyager Class ships are what you are looking for.
If you want something more like a great resort hotel, with an outdoor movie theatre, freshwater pools & spas open 24hrs, good teen & kids activites to keep them busy, basketball, mini -golf and golf simulator, ping-pong, life-size chess/checker board, video arcade, wonderful service & food then pick Princess.
Its totally up to you, you really can't go wrong with either cruise line, pick the cruise that is the best "fit" for you and your family and has the PORTS you want to visit. Ports are a huge consideration in choosing which cruise ship to take, imo.
My daughter has been on 17 cruises and she prefers Royal Caribbean ships and youth program. The youth counselors are fun and keep the kids extremely active and busy. There's more physical things to do on the Voyager class ships than Princess ships. We were just on Caribbean Princess and the basketball court is closed most of the time since it's located above the spa and it's too noisy! Not the case on Royal Caribbean...
I do agree with the above post, however, that the food is a notch better on Princess.
LEt me know what you decide.
Luisa Frey Gaynor
Family Cruise Editor