We have sailed several times on the RCCL voyager class ships and we did one cruise on the Carribean Princess. We enjoyed all the cruises and I would cruise Princess again.
If you are going on a couples trip, I recommend Princess. If you have kids then Royal is better. We sailed Princess in April 2009 and there were very few kids onboard and not many kid activities. It was a great cruise for relaxing and was quiet. We had a lot of fun and the ship was beautiful.
Royal has a lot more activities such as pool parties during the day and just a more active vibe. I have not cruised with children yet, we are booked on our first family cruise on harmony next year. However, I always remember lots of kids on Royal and lots of activities for them to do.
I found the food quality to be similar on both royal and Princess. We usually eat in the main dining room for dinner. We ate at the steakhouse on Princess and the food was excellent.
The movies under the stars on princess is fun. They have a big screen on the pool deck. They put blankets on the lounge chairs and serve popcorn and warm cookies.