I've cruised both (and I've also cruised on Premier, Carnival, Norwegian and Royal Caribbean)...
In my book, Celebrity stands out, far and away, as #1...
I was very disappointed with the entertainment on Princess (awful)...the itinerary we took (Baltic) was far inferior to the Baltic itineraries offered by Celebrity...The ship we were on (Crown Princess) was thoroughly lacking in amenities...Food availability late night was virtually nonexistent...Dining room food and service were both good, but still slightly short of Celebrity's standards...and the crew aside from the Dining Room were unhelpful and unfriendly...
Celebrity, OTOH, to my experience, has been near perfect...Food and service outstanding, ships' amenities outstanding...and the Millennium, for example, was the least crowded ship we've ever cruised on...fewer passengers per gross ton...cabins were ample, public spaces ample as well...And all the small details are handled with the utmost in class...