This is an interesting one. I see two sides to this coin:
1) He wore the Nazi uniform to a fancy-dress party. When the uniform is viewed in that context, no offense should be taken as the wearer does not necessarily endorse the ideals of the uniform, as recorded historically.
2) The prince is a permanent ambassador of the royal family and is always in the public eye, not just locally but internationally. That is the price of royalty. As such he has no immunity from acts of embarrassment, even when such immunity would usually have been afforded Joe Public at a fancy-dress.
Now, speaking for myself, I wasn't offended by it. I could have worn that uniform and there would have been no hoo-haa about it. It doesn't mean either one of us is a Nazi. But the prince should have recognised that it MAY have offended someone, and should have spared the royals this embarrassment. Unfortunately not everyone is easy-going about these matters (see the Israeli response) and the prince, in his position, has less lattitude to get away with a stunt like that than we do.