Hi Welcome to Cruise Chat. You might want to sit down with a cruise oriented Travel Agent who can point out the differences between these two cruise lines. I like both of them, but there are distinct differences. Princess is a premium cruise line and Royal is a mass market. Princess tends to draw an average age of older clientele. However; on Alaska itineraries you will see families. Along with Holland America, Princess has been established in Alaska the longest. Princess goes into Glacier Bay and Royal Caribbean doesn't. Glacier Bay is the most popular glacier for seeing calving. Royal Caribbean visits Hubbard Glacier which is also nice.
Either will be a great cruise in Alaska. If itineraries are similar but price is a huge difference, I would probably go with the lower price. But a Travel Agent can help you make the decision based on your needs and wants.