Hi Sandylynn, these are just my opinions, so please keep this in mind. I think the biggest difference is the ambience. Princess is a little more elegant without being stuffy. They also don't have announcements every 5 minutes to come spend money. Don't get me wrong, they still want you to play bingo and visit the casino and go shopping, but they just aren't on the loud speakers. Princess has the muster process down. You don't have to stand outside, and the last time we didn't even have to take our life vest. You go to your designated lounge, listen to the presentation, watch the crew put on their life jackets and then you leave. Debarkation is similar. They give you luggage tags, according to your tags you go to a lounge area and are called to debark.
Depending on your length of cruise and time of year, you may find less children on board. Although, Princess has a wonderful children's program.
For my taste, the food is better on Princess. Except they don't have the chocolate melting cake every night (drat!). The ships I've sailed on with Princess have always been spotless, the service is impeccible, and the crews have been great. Also, I've not had a bad excursion booking through them.
I've totally enjoyed all my cruises on Carnival, and every other line I've been on save one, but Princess is my favorite. So I might be a little prejudice, but these are my observations.