We've sailed to Alaska on both these ships and loved both of them.
The Volendam is newer than the Veendam and slightly larger with 3 sets of elevators, whereas the Veendam only has 2 - one forward and one aft. The Lido is larger on the Volendam, but overall you can't make a mistake choosing either of these ships.
I've only had one cruise on Holland America and it was on Veendam. We loved it. It was more expensive than Princess, but it was worth the extra money. Although I will say I still think Princess is the best value for the cruise dollar.
Given a choice between Veendam, Island Princess and Sapphire Princess, I would take Veendam, mostly because we love the smaller size ship; and the food and service was a cut above.
We did Alaska with HAL in 2000 (on the old Westerdam) - - and it was wonderful. Smaller ship with extra care and attention to detail. Our morning in Glacier Bay brought out Dutch hot cocoa and the afternoon brought out hot spit pea soup. It was such a nice touch. There was a naturalist onboard who kept up updated on what we were seeing and there was a member of the Klingit tribe giving lectures during our sea days. I've not been on Princess, but Holland America does a wondermous job with their Alaska itineraries.
I have sailed on both the Veendam (twice) and the Volendam (once), although in the Caribbean. These are two of our favorite ships. They are a little older, but are smaller and more intimate. The crew on both ships seem to be part of the fabric of the ships, making the journey much more enjoyable. I think the Volendam is a prettier ship, but both are lovely. Whichever you choose, you will not be disappointed.