I'd like to add a couple extra points to Samekichi and Eurofighter's points about the matter. I am not Muslim, so my perspective is an outside one. I'd still like to think of myself as a religiously aware person however: for that reason, I get frustrated at the blatant baiting that people do when they draw Mohammed.
When we're talking about issues like this, it's imperative not to use general terms like "Christian" and "Muslim" because there are specific sects and ideologies within these religions that promote such actions: the purest and oldest expressions of these religions do not.
What I mean by that is something like this. In Christianity, you have a split between Catholics and the many branches of Protestants. The Catholics come from a very old and continuous theological tradition that has been built and investigated constantly since days of Saint Peter. The epistles of Paul inform the writings of Jerome and Augustine; Jerome and Augustine inform the monasticism, mysticism, and other philosophical revelations of the Medieval period; these revelations inform the insights of men like St. Thomas Aquinas, who then informs the early modern thinkers like Saint Ignatius Loyola, founder of the Jesuits. The tradition has been an unbroken chain from the apostles of Christ to modern theologians like Joseph Ratzinger (Pope Benedict XVI).
In Protestantism, conversely, you have a break from this theological tradition; men like Martin Luther sought to take Christianity away from clergy and give it to the masses. Thus, individual interpretation of the Bible became more important among many Protestant sects than the religious tradition: this trend sew the seeds of what we now call moral relativism. This Protestant emphasis on personal interpretation, in addition to the benefits of its liberality, also runs the risk of allowing demagogues and manipulators to latch onto literalist interpretations of the Bible and use the to propagate hate. If a Catholic made that "Kill the F****t" game, the hierarchy of the Church would call him out on his manipulation of Christianity and maybe even excommunicate him. Because he adheres to a more relativist Christianity, he can spin the Old Testament and Gospels any way he wants without reproach from a higher authority.
Now take this lesson and apply it to Islam. The traditions of Islam stem from two major religious texts: the Quran and the Hadith. Both of these sacred texts were preserved orally for the first few centuries after the death of Mohammed and took a while to be be set in stone. While the Quran is a continuous document, the Hadith is more contested. Simply put, it is a collection of public declarations from the Prophet and Islamic law edicts (Sharia) spanning multiple centuries. The enormously complex and manuscript tradition of the Hadith means that the exact nature of Islamic Law is somewhat… open to interpretation.
Over the last several centuries, the clerics of Islam spent lifetimes learning the Hadith and advocating from different theological schools how to apply the Hadith appropriately to everyday life among Muslim peoples. Interpretations differ among the Sunni and Shi'a most drastically, but there are subdivisions even among these two sects that have further interpretation differences.
Enter Wahhabism. Al-Wahhab was an early modern theologian from the Arabian peninsula who argued a literalism interpretation of Islamic Law, meaning every word on the page had to be followed: no metaphors, no symbolism, no allegories: the words on the page in his sect's manuscripts of the Quran and Hadith needed to be followed to the letter. I brought up Protestantism earlier because this is like the Creationist Approach (Catholics cannot be Creationists): because Genesis says the Earth was made in seven days, to them it was made in seven days: no way that could have been a metaphor to them. Thus, if a Wahhabi sees "kill people for depicting Mohammed" somewhere in the Islamic religious/legal texts, then they'll use that passage to justify their killings even if that passage was part of a metaphor or something like that.
The Ottomans and Qajar Persians, the two dominant Muslim nations of Al-Wahhab's time, both viewed this Wahhabi ideology as extreme at the least and heretical at most. For a couple centuries after Al-Wahhab, the ideology remained small and niche.
Then, the House of Saud came to dominate the Arabian Peninsula and along the way, they took up the Wahhabi doctrine as their state religion. With the insane amount of money flowing into their country from international oil trade and the renting of land for business, the Wahhabi House of Saud dedicated some of their wealth to building Wahhabi mosques and propagating Wahhabi literature throughout the world.
When the USSR invaded Afghanistan, it wasn't just the USA who sold them arms and provided them with military training; while the USA just provided the hardware and political support, the Saudis brought the Mujahideen their Wahhabi ideology as well. Fighters from Saudi Arabia also went to volunteer and help out in the fight against communism and when the Soviets were beaten, they left with Afghan notions of militancy. Among these fighters was a particularly evil man, none other than Bin Laden himself.
The Wahhabi extremists of Saudi Arabia have funded and provided the ideological basis for groups like Al-Quaeda and ISIS; these extremists essentially stem from the most extreme logical result of Wahhabism. Furthermore, while traditional Islam adheres to the wisdom of (mostly apolitical as I understand) lifelong clerics, many foundational Wahhabi thinkers were engineers and other STEM professionals who then brought their literalism to religion, depriving it of abstract concepts like mercy and pursuit of the good and making it all about brutal adherence to the laws.
These terrorist groups basically use misinterpreted selections of the Islamic religious/legal tradition to excuse their terror tactics as they attempt to carve out a larger sphere of influence for themselves. The reason why most of the rest of the Muslim community doesn't fall in with this thinking is because they mainly aren't textual literalists and have more interest in living their own lives in pursuit of the good rather than forcing their religion on others.
Again, this is the opinion of a third party. I consider myself Christian with a bias in favor of Catholicism and I have a very limited education about the Muslim world, so this is by no means the assessment of an expert. I'd appreciate any feedback from practicing Muslims if I've gotten anything wrong.
