I remember being at Disney with family, including an 11 year old and 14 year old. The youngest wanted to stay in the room, get room service and watch tv. The older one complained that it was really lame to be in Disney for 10 whole days!! As for the Cape, June is early in the season and it's apt to be cool. You guessed it, I vote for Aruba!
For me it all depends on the season. In the summery months, we love to go to the Jersey shore since its a hour and 15 minute drive, and no flight is needed. Fall, Spring, and Winter then Aruba wins. When you have kids, they love the beach in day, and the board walk and rides at night. Retirement plan:
Aruba: Dec 26 - Easter
Jersey Shore: Rest of the time