I've been to Aruba (stayed at the Marriott Surf club) and will be heading to Grand Cayman later this month. I did not have anything stolen or any problems with crime when I was there. If no one else offers a comparison then I will try to remember to do one for you once I get back from GCM.
One thing to consider is that June is start of hurricane season and Aruba is generally out of the path of most Atlantic hurricanes while Grand Cayman gets hit with much higher frequency.