I wonder if it's a generational thing? For a twenty year old, the holocaust, however dreadful, is history. And the Nazis do have a camp funniness, which is why Mel Brooks can do Springtime for Hitler and it be funny. Perhaps the Nazi regalia just doesn't have the same gut response from their generation as it does for older generations?
Having said that, these are the excuses I would make for a normal 20 year old. Someone like Prince Harry, who is a member of the Windsor family with all that means, would surely have a greater consciousness of just how wearing a full Nazi get-up would be perceived by large sections of the media and the public?
And where exactly did he get the costume from? Your average fancy dress shop doesn't usually stock full Nazi Stormtrooper regalia, does it? That's what makes it stink for me - whoever procured it for him, knows some very dodgy people.