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We have cruised both and found them excellent. I prefer the dining arrangements on NCL, as this allows you to dine with different people each night of the cruise.
Have never travelled with Celebrity but have with NCL. They do have formal nights like any other cruise line where you can dress up. The rest of the time is how you feel about dressing up.
You can either eat at set meal times or opt to go 'anytime' dining. With that you can either eat in the buffet or there are one or two dining rooms which you can eat in. A lot of people tend to reserve times for these but when we have done it we have just turned up at the door and been seated immediately.
We have travelled with both and prefer NCL you have the freedom to dine whether it is in the main dining room or in one of the speciality resturants or the buffet there is no need to book sitting times. We also found the dress to be very similar on both, we also found the entertainment much better, celebrity is more reserved.
I have found that not many people dress up these days on NCL, not compared with other cruise lines, where I have found about 90% of the passengers do make an effort and dress formally.
Have done both. The main difference is the Anytime dining (which I love) - go to dinner when you want in the main dining rooms, or one of the alternative restaurants. Otherwise prebook one of the speciality (teppanyaki, French, Steak). They have all the main dining room menus on display so you can see what night does not appeal and then book elsewhere. A good idea, as on X you are taking a chance on what meal you are missing! Otherwise the decor is one thing that you will notice - NCL is rather garish. Expect a different experience - and try the macaroni cheese in the snack bar area one lunch or the fish and chips - it is delicious.