Ihave done 25 cruises, 5 with Carnival, 2 with Princess and 2 more booked with Princess. By far, Princess is the classier line. But having said that, some people love the neon and "carnival" atmosphere of Carnival and may find Princess too subdued.
Princess' prices are not as competitive as Carnival but I'm more than willing to pay for the ambiance of Princess.
Ditto to everything above with these additional notes:
I think Princess has a few more add ons than Carnival. I found it bothersome that the steam room on the Island Princess last year was an additional fee. But it was a dynamite experience - not your run of the mill steam room.
There's the ice cream thing which has already been mentioned.
Carnival's past passenger program gets you nothing. Maybe some discounts on selected cruisings but that's all (and I'm not sure that's even truly a discount). Princess allows you to accrue points for referrals of friends, which get you a cash discount on your next cruise. Along with some of the usual type perks. You can see them on the Princess.com page under Captain's Circle.
Princess does the best job of anyone in Alaska.
The layout of their cabins provides more privacy even though the actual space may be a bit smaller.
Personal Choice dining is available as well as traditional. Nice to have a choice.
Personal enrichment classes are available on board. On the Island it was pottery throwing. I've forgotten what it is on the Grand.
All in all, Princess does a superior job to Carnival, although I would not hesitate to sail Carnival again if the price was right.