The essence of any successful interaction between people of different cultures is RESPECT and at the heart of respect is 'sensitivity' to others, their feelings and their culture. If we wish respect for what we hold dear and cherish, unless we are prepared to give it, we cannot expect it in return. To make fun of someone or to derogate them because of their colour, culture or beliefs shows no respect at all for who they are and wish to be. Though words in themselves are harmless, it's what they have come to symbolise that causes offence.
For example, the terms 'Paki' and 'Raghead' might not matter to people of low esteem who need to put others down to feel superior and powerful, but they are highly pejorative terms mainly used by white people to describe people from the Indian sub-continent and Middle East. The term 'Paki' was deliberately coined to put down people from Pakistan, even when they are NOT from Pakistan, which is what makes the word so offensive. It is like calling a Black person 'n-word', so it isn't very nice. It emphasises primarily the colour and perceived inferiority of the person referred to, and the xenophobic, fearful and disrespectful tendencies of the person using the term. Why would anyone want to call another person a 'Raghead' just because they wear a turban or they are an Arab? Why is it that some privileged people are free to be what they are and others are not? Mainly because of ignorance, innate prejudice and fear among people with the power to deny others their heritage.
By using those terms, Prince Harry was exhibiting racist behaviour, plain and simple. But is he personally racist?
One could say that, coupled with the insensitive Nazi uniform he wore, he was being racist. However, the key question is whether he was being maliciously racist (ie: knowing better, but still doing it) or whether his racism comes through ignorance and he simply needs more education about other cultures and diversity. However, the fact that Prince Harry even made fun of his own grandmother (the Queen) to his colleagues does not really suggest a racist person. It suggests one of low self-esteem seeking attention and approval from others by using any means he knows to ingratiate himself with them, even to put his own family down to become 'acceptable'. In our British parlance, vainly trying to be 'one of the lads' and failing miserably because of the expectations around his own unique heritage.
But we shouldn't blame Prince Harry too much for his behaviour. He is simply a branch of the big oak tree. One has to dig deeper for the root cause of his disrespect. And we won't even mention the racist gaffes by his grandfather, Prince Philip. Look closely at the royal family, especially at the Queen who is head of a multicultured Commonwealth, and Prince Charles who is next in line to the throne. When did we ever see anyone Black/Asian or whatever around them in Britain? Think about it? Any minority staff, minority advisers, minority friends, minority ladies-in-waiting, minority private secretaries? Are there any at all? Quite simply, nope.
What we have always had is a monarchy that proudly heads the Commonwealth, fraternising regularly with the white sections of it, while keeping well away from interacting on a personal level with its people of colour. No wonder there is fertile ground for racist behaviour and cultural disrespect!