Hi Joe Cook! How's it going?
The first thing to say is that the issue is "satire." Its important to keep that firmly in mind. The Danish publication a few years back, and no doubt the makers of the latest film... what is it, "The Innocence of Muslims?" no doubt see themselves as satirists. But what is satire?
Satire is NOT just making fun of people, cultures, or institutions. Satire is "making fun" (for lack of a better term) the powerful in society. If you make fun of people who are not powerful then your just being mean, and, frankly, something of a bully.
For example, in the United States in the 1920s, you had white performers like Al Jolson who got on stage in "black face," as they made their living imitating black people. Now, this was NOT satire because there wasn't anything like parity between black people and white people at that time: black people were still being lynched in the south and Jim Crow segregation laws were in effect, and so forth. So, what you had there, was a powerful group (whites) being mean in making fun of a comparatively powerless group (blacks).
So, I would say making fun of your UK Royal Family would be satire because they are on the upper tier of society; I would imagine they are powerful, even if their nominal role is ceremonial. The Arab-Islamic world is not on parity with the West in terms of power. After all, it is in Arab Muslim lands were Obama's devastating War on Terror are being prosecuted with all the drone strikes, and so forth.
Pakistan, Egypt, Afghanistan, Palestine, the Sudan, Iran (whom we seem to threaten to invade every other day), and other countries cannot hit "us" back with anything even mildly approaching such ferocity. The Arab Muslim world is far, far, far, far WEAKER than "us" industrially, militarily, and economically. So the cartoons and the like about Prophet Muhammad are being MEAN to people who are, at this point in history, comparatively powerless.
Look, it would be like making fun of a triple amputee, who, perhaps lost his limbs in the Army or something, and saying to him, "Hey there, stumpy!"