The public are showing their support for Prince Harry because he is the only one of them who appears to be human most of the time.
Out of the direct line to the throne, Prince Harry is third after his father and older brother and of course those two keep a rather tight leash on their appearances. You never seem to hear of them doing anything silly or dumb, or see them as people.
So when Prince Harry is caught smoking we smile, even when its weed, when he is caught eyeing up some ladies we smile, when he does the Bolt pose with the great man himself we smile, when he's caught drunk we smile. So you know, him naked, acting like a lot of party going 20-something people do, well a lot of us just shake our heads with a smile and say "Thats Prince Harry."
I've yet to meet a person irl who found his actions to be gross or disrespectful to his duties.
Certainly everything he does after wearing that nazi costume is ok in our eyes because he's NOT done that again.
He's just a bloke who happens to be a prince.