I've been on both ships in the last couple years. Both cruise experiences were enjoyable, but for our preferences, especially for Alaska, I believe I'd choose the Star. Like Parker, we enjoy the freestyle dining concept, and how it works - and my husband does not like to dress up! Princess will be somewhat dressier on the formal nights; NCL will have a whole range of dress. Princess does have "Personal Choice" dining, allowing you to choose open seating instead of being assigned a dining time and table.
On the other hand, Princess has been doing Alaska for many years, and has invested rather heavily in the destination, building its own lodges used for cruisetours. The balcony cabins are larger and, IMO, nicer on Golden Princess, although the bathrooms on the Star were large, and the shower was positively luxurious by cruise ship standards! Food is, for us, a wash; my personal preferences don't match Princess food, although in our experience the quality of the food & service in the main dining rooms is superior to NCL's. The Star has more alternate restaurants of different types (most extra charge, but worth it); the steakhouse on Golden Princess was absolutely outstanding, and the one on the Star was close to as good.
I've posted reviews of both ships on cruisereviews.com, our sister site. Perhaps those will help (although different itineraries, the ships are the same as the ones you are considering).