So, the four portrayal problems...
- Religious Taboos: Some Muslims (far from all) consider it haram (forbidden for believers) to portray the prophet at all, including text.
- Freedom of Speech issues: In the Middle East as well as in The West.
- Cultural bullying issues: A powerful group (in this case Westerners/Christians) joking about something important to another group (in this case Muhammed to Muslims) can be perceived as bullying.
- Hate-speech issues: The previous problem is made far more infected by Western culture's history of hate-speech and crusades against Mohammed/Islam/Muslims note
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?).
The other thing which might interest people of other Abrahamic faiths is that the aniconic non-representational attitude in Islam is actually present in the teachings of the Judeo-Christian Bible itself. The Catholic Church from its early years had a number of fights with dissenting sects on portrayals and depictions of Jesus, as well as representation of certain icons, and not portraying Jesus or the divine was actually official policy in the Byzantine Empire for a time, creating a serious argument within Orthodox Christianity. So the tradition of Islam in being against representation is not really something unique or weird to themselves, but in many ways Truer to the Text than Christianity which specifically provided loopholes to develop and patronize art on Christian subjects during The Renaissance, which was protested by the otherwise quite conservative Martin Luther.
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/governments use religious controversies as an excuse to try to deny the citizens of their countries access to the Internet.
There's also a violent sizeable Vocal Minority (keyword still being minority) with excessively vile views and they're not afraid to shout these on social media every time a terror attack happens in a Western country, especially if the victims are in part, or entirely, LGBT. This, in combination with the aforementioned religious dogmatism and extremism, leads to the conclusion or not that Islam's values are incompatible with those of The West... which loops back to the issue of portraying Islam/Muhammad, because someone will be offended by it and again start some form of violence.
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.
It's also a point that most of the time, the portrayal and depictions of the Prophet by Western artists and cartoonists often lead to criticism on the part of most Muslims. Most of the time, the understanding of the Prophet and his life is not given careful reading nor do they pay attention and refer to Islamic depictions in the past. That is to say, it is very rare for an American/European non-Muslim artist to depict Muslim culture that most people who live there can recognize and take seriously.