So, the four portrayal problems...
Religious TaboosA lot of Muslims consider it to be haram (forbidden) to describe the prophet. This include words and texts as well as images, and is not limited to negative portrayals. Of course, this varies a lot, as different groups of deeply religious Muslims have different views on this. Also, a lot of Muslims do in fact not want their own lives (much less the lives of others) to be ruled by religious taboos. At the core, it's a matter of whether or not we want religious scripture to dictate the rules for public secular life - the same kind of issue as whether or not The Bible is a valid reference point for deciding whether or not homosexuals should be permitted to get married in civil unions (which unlike a church wedding is a strictly secular matter). Religious leaders are fighting very hard to make themselves relevant by claiming a right to decide what people should and shouldn't be allowed to do. It should be noted that the issue of portraying the prophet or not is mostly a debate within Islam, rather then a debate with pious Muslims and their supporters versus liberal Muslims, atheists of all nationalities and religious people of other faiths then Islam. One widespread belief within Islam is that one ought to be like the prophet, and in some interpretations this includes that men ought to try to look like the prophet. Thus, portraits of Muhammad is an entire genre of Islamic literature and art. Banning portrayals of Muhammad is traditionally mostly a counter-reaction against this. Also, in certain (very extreme) branches of Islam, the taboo is not about Muhammad. It's about portrayals, period. It's forbidden to paint pictures or (in the most extreme—and thus little-followed—branches) write stories, no matter what one is painting or writing about. Such extreme views on the matter led to the Taliban attempting to ban television in the years they ruled Afghanistan, as it necessarily involved "portraying" people. The act of creation belongs to Allah alone, and trying to create works of one's own is blasphemous hubris. What "creation" is is another story, one that those who accept this idea tend to argue about ad infinitum (some say painting is unacceptable but photography is OK, some say both are bad; some say you can paint, but you can't paint animals; some say you can paint animals but not people; some say you can paint people and animals but not human faces; some think music is unacceptable, while others don't;note some say that all kinds of creative writing are acceptable, while some say that it's okay if it's nonfiction but not fiction, and others ban it entirely; etc.; etc.; etc. Who said that it was just Jews who liked to argue?).
Freedom Of Speech issuesIn the western democracies, authors of fiction had to fight long and hard for the right to write about religious themes. In most of the Islamic world, the struggle for basic human rights and basic freedom of speech is still being fought. While more and more Islamic countries are gradually becoming more and more democratic, there's still a lot to do. Undemocratic forces use religious controversies as an excuse to try to deny the citizens of their countries access to the Internet.
The bullying issueAdd to this already complex situation that while Christianity is deeply rooted in Western culture (and to some extent in Japanese culture) Islam is not. Thus, when Western works poke fun at Christianity, deconstruct it, play around with it or whatever, they are doing it to their own culture. When Western works play around with Islam, it is instead perceived as stepping on another group. One that is often perceived as oppressed or at least less powerful.
The hate speech issueThe 2000s saw an upsurge of anti-Muslim feeling in Europe, North America and Australasia related to the 9/11 attacks and smaller-scale terrorist attacks by Islamist extremists, but also with strong implications that criticism of Islam was being used as camouflage for straightforward racist prejudice against ethnic minorities, given that most Muslims are not white. On the other foot, you have the faithful accusing any criticism of their faith as being "racist" even though criticism of a religion itself is not inherently racist, not even when the subject is Judaism.