I have done both cruise lines and believe that each has its benefits.
Carnival - In my experience (two 7 nights and one 8 night), it is a rowdier atmosphere. However, I (early 60s) generally find it just fine. I like the Carnival comedy club and think they have the best games. (I love games.) Negative, the food, which I find mediocre at best. But food is subjective and everyone has their own opinion and no one is wrong there. I can always find something I like. Always. Number of kids depends on when you go. I think the cabins are nice on Carnival. I have another Carnival booked.
Princess - In my experience (17 or 18 cruises), it is more sedate. Not as formal as it used to be, which is just fine with me. It is fun, and I always have a great time on Princess. The entertainment is also good on Princess. Food is much better, IMHO (see above). Negative, they don't have a comedy club and their games, while very fun, generally, aren't quite as much fun as Carnival. Cabins are just fine on Princess. I have another Princess booked.
My favorite cruise line is Celebrity. All else being equal, I'd pick Celebrity. Reason, I think Celebrity is just a bit nicer, all the way around. I don't have a Celebrity cruise booked right now, but I will at some point.
I agree with Kwokpot that it all depends on what you are looking for. If you want something more informal, I'd certainly recommend giving Carnival a try. Just my thoughts.
Ohiodoglover