I've cruised both and love both. The Princess ships are larger, have more venues and are a little more elegant. The Holland America ships carry fewer passengers, are more reserved, have better food and service (in my opinion--and I'm not saying there is anything at all wrong with Princess' cuisine or service), and have a more intimate small-ship feel.
I think I would recommend Princess for a first cruise; but either will give you a wonderful experience. You should talk with a cruise-oriented travel agent, who can tell you more about how the two lines and ships compare.
One other consideration: the Holland America ships you mention don't have any moderately priced balcony cabins; they jump right from "oveanview" to "Verandah Suites". The Princess shps have hundreds of affordable balconies, and a balcony is really nice to have, especially in Alaska.