In addition to what PCBoston said, I'll add the following...
Princess does a better job on their shore excursions.
Princess and Carnival both have excellent children's centers, but I think Princess has the edge on activities for older teens.
Carnival has 24-hour softserve "ice cream" but Princess has real homemade ice cream. Problem is Princess only has that ice cream 1) in the buffet at 3:30 in the afternoon for an hour or less, or 2) with dinner in the dining room.
Princess has Anytime Dining in addition to fixed times, Carnival only has fixed times.
Carnival has better adult comedians in the late night shows.
Princess has better art auctions.
You can read further details on my web site at www.MyCruiseAdventures.com
where I've sailed Carnival twice and Princess twice.