My wife and I took her mother on her first (our third) cruise for Christmas. Our first two were on Carnival - one of which was our honeymoon last year. This was our first Norwegian cruise. We compared NCL and Carnival the entire trip:
Embarkation/Debarkation: NCL wins. Seamless and easy. It took at most 10 minutes to embark. No lines to debark. We used the "normal" debarkation (i.e. leave your bag outside the door). It was also very easy to get on/off at each port of call. They even have staff outside the ship at each port with water & hot towels to wash off and hand sanitizer when you get back on. Mother-in-law was particularly impressed.
Entertainment: NCL. No comparison. The shows were much better and entertainers are much more talented than Carnival. There seemed to be more crowd involvement in the various activities (which we enjoyed more than Carnival). Many activities seemed more original (Dancing with the Stars, the Quest, and Liars Club to name a few).
We liked how the cruise director only came on 2-3 times/day to remind of activities, compared to once/twice an hour on Carnival. We think Carnival tries too hard to be "fun" and constantly reminds you of all the "fun activities" going on. NCL relies on you to read the daily activities and figure it out on your own.
Food: Carnival. All of our meals on the NCL Sky were great. But Carnival has the Chocolate Melting Cake. The freestyle dining is great. We did not eat in any specialty restaurants. Drinks are pretty pricey @ ~$7 each.
Ship: It's a wash. Carnival's ships flowed a little better, and were a little more open than NCL. But NCL has more quaint little lounges/bars for their activities. NCL's decor is more updated than Carnival too. We loved the Dazzles lounge and Captain Cook's lounge. (Dwight was great on the piano with tribute nights to Billy Joel and Elton John).
Staff: Both cruise lines have very friendly staff who will do anything you ask of them. We liked how the NCL waitstaff wasn't as pushy when it comes to selling drinks as Carnival is. My Mother-in-Law was amazed at how friendly everyone was (she herself is extremely friendly). Cute towel animals.
Customers: NCL. Customers seemed a bit more upscale/friendlier on NCL. We got the impression that Carnival is good for beginners, and NCL is great for repeat cruisers. In fact, out of every group/couple we spoke to, at least 1 person had cruised before (not necessarily NCL). People keep to themselves more on NCL, except in the activities, where there is much more involvement. The crowd is slightly older on NCL than Carnival...not by much though.
TLDR; We probably prefer NCL over Carnival, but each has things we prefer here or there. NCL is a party boat, but not as much as Carnival. If you've cruised before, I think you'd really enjoy NCL. We'd have a hard time choosing between the two.