Comparison between Princess and Celebrity Cruise Lines
♦ Routes (free): Detailed travel itineraries in USA
General
There are differences between the various cruise lines in service levels, food, shows, what is included and what is not, density, crowding, and more. It is somewhat difficult to compare companies because there are also differences between different ships of the same company, and even on the same ship, changes occur over time. Comparison is easier when taking two (or more) consecutive cruises on ships of different companies, as there is no time perspective gap or different seasonal effects. In winter, there is nothing like the Caribbean for a couple's or family vacation, especially for lovers of exotic beaches and islands, summer enthusiasts, and cold-haters like me... I spent a week cruising on the Celebrity Beyond of the Celebrity Cruises company, where I had already taken a long cruise in the past, and a few days later, a cruise on the new Star Princess.
The ship Celebrity Beyond:

The ship Star Princess:

And about a year ago, I cruised on the Sun Princess from the same series.
The pool near the "Dome" on the ship Sun Princess:

For dessert, I sailed on the new Celebrity Xcel, from the same series as Beyond but with several innovations and surprises. The comparison here refers mainly to these ships.
The deck on the ship Celebrity Xcel:

These ships are from the newest series in each company and are all recommended. Nevertheless, each company, especially within the same ship series, has advantages and disadvantages. It is important to emphasize that both companies are considered premium lines and are at a higher level than Royal Caribbean, Norwegian, or Carnival, and of course much higher than the Italian companies.
Shows
The shows on Celebrity are at a higher level; the theater is much larger, impressive, and comfortable. There are also many more large productions by the ship's entertainment teams.
The theater on Celebrity Xcel:

At the same time, Princess has more activities, games, and dancing occurring simultaneously. On Princess, a celebration is held with champagne waterfalls while the captain introduces himself and the heads of departments.
Star Princess - The Captain and department heads and a champagne waterfall at the Princess Plaza

In Celebrity, this does not exist (there is an introduction of department heads before one of the main shows in the theater), but the captains in Celebrity are much more present, friendly, and accessible to the passengers.
The Captain in a show at the theater of Celebrity Xcel:

The Captain and department heads and many seniors in Celebrity are present at the theater entrance, in the central piazza, and on the decks, and some of them participate actively in various events and games. Princess blows its whistle upon departing every port with the opening notes of the theme song from the legendary series "The Love Boat" (which was filmed on an ancient Princess ship), which warms my heart every single time. Celebrity has a regular and standard whistle like most companies.
The Rooms
The rooms in Celebrity—where in both cases we are talking about standard balcony rooms—are more spacious, arranged more comfortably, and most importantly: the bathroom in Celebrity is much more spacious.
The room with an infinity balcony on Celebrity Beyond:

In Princess, it is a real balcony, and in Celebrity, it is an "Infinity Veranda"—a room with a huge window that opens like a balcony, which enlarges the room but you feel less "outdoor." Some like it very much, and some really don't... In both companies, rooms are cleaned and arranged twice a day.
The room with a balcony on Star Princess:

Room service (for food and drinks) on Princess is included in the packages, while on Celebrity it is always at an additional charge. On Celebrity, a chocolate still waits on the pillow every night for every passenger, which used to be customary in all companies and was canceled, and Princess has already canceled it as well.
The Decks - Pools and Walking Tracks
The walking tracks on Celebrity are longer and much more recommended.
The walking/running track on Celebrity Beyond:

The pools/jacuzzis/gyms are at a very similar level. In both companies, there are towels for the pool and the sea freely available (without the need for registration, without the need to return a towel to take a new one as is the case in lower-level companies).
Food
The food is at a similar level, but Celebrity has more dining room options included in the price. Specialty restaurants at an additional cost—plenty of good options in both companies. On Princess, it is usually cheaper. Both companies have different food and beverage packages. Princess's is much more worthwhile and includes much more. On Princess, the desserts are more elaborate and the variety is larger.
Desserts on Star Princess:

But for liquor lovers—there are very few sommeliers on Princess (pourers whose exclusive role it is), and most of the time the waiters and assistant waiters bring the requested wines in glasses without pouring from the bottle and giving a taste. Both companies have excellent coffee shops. Princess's is active 24 hours a day and includes, besides cakes and desserts, various and diverse savory pastries. On Celebrity, the coffee shop is active many hours but not at night, and it has very few savory options and sometimes none at all. But: on Celebrity, the coffee shop is located in the heart of the central "Piazza" with plenty of soft and pampered armchairs and you can almost always see and hear good music from there, while on Princess there are few seating places adjacent to the coffee shop.
The cafeteria on Sun Princess:

However, there are a variety of options nearby, both indoors and on the open deck. On Celebrity, waiters no longer offer ground pepper on every dish during meals, while on Princess the custom still exists. When returning to the ship from a port excursion, both companies pamper the passengers with cold towels and cold water. The feeling, regardless of how much you enjoyed the port and the tours, is: how fun, we've returned "home" to the pampering and the wonderful personal treatment.
Laundry
A significant advantage for Princess in my eyes: they have self-service laundries with washing machines and dryers on every floor. This is, of course, in addition to the very expensive laundry service provided by the cruise line (which is the only option on Celebrity).
Room Key / Medallion
On Princess, instead of a key, there is a "Medallion"—it is much smarter than the key because it senses the passenger approaching and unlocks the door. Even if you arrive with full hands, there is no need to put everything down and fumble with the keycard. The Medallion comes with a neck lanyard, and there is no need for pockets or to specially buy a key carrier.
Elevators and Access
Star Princess has three sets of stairs and elevators: in the forward third, in the center, and in the aft third, which is very convenient compared to Celebrity Beyond, Xcel, and the entire Edge series—they only have two sets of stairs and elevators, and whoever lives in the center of the ship has to walk long distances through the corridors to reach one of the stairwells. At the same time, the elevators on Celebrity usually arrive much faster than on Princess, regardless of whether it's rush hour or a quiet time. The main dining room on Princess, located on Deck 6, can only be reached by the aft elevators. Those living in the center or forward are forced to go down to Deck 7, walk to the aft, and then go down a deck, which is both tedious and confusing for people without a sense of direction. In general, access on Princess to public areas and the buffet on Deck 9, which is the central deck (where the coffee shop, restaurants, future cruise booking, and tours are also located), is not pleasant for those coming from the bow. You are forced to walk along long corridors near staterooms (a very bad floor to live on, despite the luxury "Veranda" rooms located there). Access to the main guest services office is also problematic and through stateroom areas.
Loyalty Program
Both companies have loyalty programs that are not particularly attractive. On Princess, the highest status, "Elite," does not really justify the loyalty. On Celebrity, the similar status is also called Elite and the benefits are similar and not exciting, but there are two more "tiers" above it, with the highest one called "Zenith" requiring you to almost live on the ship for most months of the year... In any case, there is no justification for sticking to one company and sailing only on it; no benefit you receive justifies that.
Summary
Each company has advantages and disadvantages that most people wouldn't even notice. Both are wonderful companies and the ships I tested are the best and newest of each company. In any case, I highly recommend not doing two or three consecutive cruises on the same ship (what is known as Back to Back) but rather diversifying: when sailing two consecutive cruises on the same ship, the shows are identical on the different cruises, the menus repeat themselves, and so do the various activities. When diversifying, you get to see and do much more, to innovate and experience.