Holland American and Princess are actually owned by the same people (now), but the culture is different.
Holland America is normally considered, by cruising standards, to be a bit more formal, usually appealing to a bit of an older crowd.
Princess caters to a younger set, and has all the differences that normally come with that.
My sister lives in Anchorage, and used to work for Holland America on one of their trains that goes back and forth between Anchorage and Denali National Park. She quit a few years ago to pursue other things, but enjoyed what she did.
When my mom retired, she went to work for Princess, also in Alaska. She spends every summer at the Mt. McKinley Princess Wilderness Lodge near Talkeetna. Cruisers often book a land/sea package, and the lodge is the "land" part of the deal. She loves it, and she loves Alaska.
I haven't been yet, and am so jealous of anyone that is going. But happy for you, too! You'll love it, I'm sure.