Two totally different trips. We liked the both for very different reasons. Alaska is more of a sightseeing trip. There are plenty of adventures too. It is not about relaxing by the pool our sunning on the deck. It is much more about engaging with your surroundings. It is awe inspiring.
Caribbean has more what I would call lazy time. There are still things to see and do, but there really is more down time. Even the sea day in Alaska offers plenty of unique sights and surroundings as you are never really far from land. The sea days in the Caribbean are more about the ship and the onboard activities.