I'm a teenager and i've been on numerous Royal Caribbean & Princess cruises. I can assure you that the Crown Princess has a alot to offer for kids and teenagers. Princess has 3 groups for kids: toddlers, kids, and teenagers. Everyday, Princess places a patter outside your room explaining the list of activites for that day. I found this very useful. SOme things that are offered are: movies under the stars, games, a live DJ, & anything that allow kids to meet others on board. Not only are the clubs decorated appropriately, the staff is very helpful and generous.
The reason I would choose Princess over Royal Caribbean is because although Royal offers amentities such as a ice skating rink, climbing wall, etc, the possibility of using these (without waiting in line) is very low. Royal Caribbean is a wonderful cruise line, although I found that I had more fun on Princess.
So if I had to choose, I would choose Princess without a doubt. Also, the Crown Princess is a beautiful ship, with 600+ kids (usually). They have alot to do at night as well, such as movies under the stars, dancing, and trivia. Overall its a kid friendly ship.
Im not sure how strict you are with your kids about curfew, but on the Royal Caribbean ship, kids (under 18) cannot be out passed 2 am. This is for safety issues, although I would much rather be able to do what I please throughout the night. Princess & Royal Caribbean both have outstanding security.
Hope this was helpful!
Good luck on choosing!