I just sailed Carnival and I have to give the edge to Princess in food.
For instance:
I have the wonderful Princess Fettacunni almost every night as a side dish. One night Carnival offered this and there was no comparsion. The Princess offering is creamy and wonderful, the Carnival offering looked like cream of mushroom soup poured over noodles. Even DH, who thinks I am too critical sometimes, thought the same about the food. If I remember correctly Princess changes up the menu a bit at breakfast in the dining room. Carnival did not, it was the same offerings every day.
It seemed like Carnival had more home type foods, such as BBQ ribs one night, short ribs another night. We look for more interesting food choices when we travel. I would tend to think Princess probably has a larger budget for food than Carnival. I would think the higher the fare the more budgeted per passenger for food.
Now, that is not to say we went hungry on Carnival because we did not. There were lots of great dishes. That warm chocolate melting cake was heaven on a plate. I also enjoyed bagels, cream cheese and salmon almost every morning while DH had eggs benedict. DH also liked the tuna fish sandwich from the deli. I had a roast beef and brie one day from room service that was very good.