Having sailed with Princess primarily, in comparison to Celebrity these are the highs and lows:
Newer ship (solicitized in Feb 2012)
Entertainment; shows lasted longer than the standard 45-minutes on Princess, and were well-done
Martini bar; way cool!
Drink package; freedom from sticker-shock/worry
Fitness; no free-weights, or cable machines. $20 day/$99 per week for access to steam room
Buffets; slow, lack of choice, and real lack of availability from 2-5 (right when you're getting back on the ship following excursions or trips ashore). This sailing only had pasta and salad from 2-5. Hamburgers and Hot Dogs were available poolside.
Main Dining Room; tables all over were quite small; there was no place to move, and I didn't get bread at dinner because the plates were too crowded. Dining Room was also quiet noisy
Room Service; horrible, had breakfast several times, and each time major mistakes; entree's forgotten, no silverware, etc. Lack of menu items
Ship Staff; Lazy. Cabin Steward left 2 rollaway beds on the balcony for the entire cruise; when asked about them he explained that they were difficult to move. Frequently out of ice, Never picked up trash on balcony that had blown down from above.
Cleanliness; This ship seemed dirty; there were always fingerprints on banisters. In the buffets many plates (sometimes 5 or 6 in a row) that were supposed to be clean, had food stuck to them, were still wet, etc. Trays were dirty, glasses were dirty. With all the Noro Virus concerns, I would have hoped to see higher temperatures on the plates, and more checks on food holding temperatures/times. I never witnessed this on Summit, but have seen it regularly on Princess.
This was my first cruise with Celebrity. We arrived at the pier at 2pm, to find a very long line. Because we had a Concierge Class room, we were able to bypass this and get to the shorter CClass line. Our bags also received priority and arrived within an hour of boarding the ship. The public areas of the ship are nice, and it is obvious that the ship was recently refurbished. This was my first Aft-Facing balcony. The room was great; this was a last minute booking so Celebrity waived the handicapped requirement, and booked the room. It is quite large; about 300 sq. ft., and the balcony is approximately 250 sq. ft. Noise from the restaurant was noticeable when standing at the railing, but underneath the canopy it was not noticeable. We did have some trash accumulate over the sailing; mostly straw wrappers, and one cigarette. The bed and pillows were comfortable, better than Princess. Room furnishings were new. Concierge Class does not seem to be worth the extra money, except for the aft balcony which was outstanding. Location was also good with rooms above and below, yet close to dining rooms, the sunset bar, and fairly close to the pool. In the future, I would not book Cclass unless it was aft-facing. The evening snacks were the equivalent of toast with cheese-whiz and some smelly shrimp. We opted to request a fruit and cheese tray. Something any cabin could request through room service.
Around the ship, the staff wasn't very outgoing. The conversations were frequently awkward, and disjointed. There was no service anticipation (of needs); something I truly missed coming from Princess. The bar staff was good in the Martini Bar, and Sunset Bar at the back of the ship.
The fellow passengers on Celebrity seem to be a bit younger than Princess, but everyone I ran across was friendly and well-spoken.
In the future I would book either Celebrity or Princess, depending on price, itinerary, and date/cabin availability. If all those were equal, I'd choose Princess. While the ships aren't as new, the service, food selection, and cleanliness lend to a better overall experience.