Take JC out of the equation, and that dove and what are we left with - two religions that worship the same god, so I have to ask - whats the bloody problem?
Islam differs from Christianity in being a return to an older-style Semitic religious legalism. Except in Islam, the legalism isn't intended for the Jewish people alone, designed to preserve their ritual purity and to set Jews apart from others as God's priestly nation.
In Islam, the legalism is supposed to be for everyone, a divinely-revealed and in its fundamentals unchangeable social order that's supposed to be extended to the entire human race.
I guess that the Jews and Christians, the "people's of the book", recipients of earlier prophetic revelations from the same God, can be allowed to worship that God in their own way within the Islamic social order. But only so long as Muslims have authority over them. Religions like Hinduism or Buddhism are less lucky, not being worshippers of the same God. So in theory, the only option members of these other religions are given is conversion to Islam or death.
Of course in real life, that isn't always going to be practical. So when the Muslims conquered India, the Hindus were generally treated as dhimmis, as if they were people of the book under Islamic rule, even though they technically weren't recipients of what the Muslims accepted as earlier prophetic revelation.
That was a big point of controversy in medieval Indiam Islam, with religious legal scholars periodically calling for the conversion-or-death ultimatum to be aggressively enforced against the heathen idolators. And that history in turn explains some of the religio-political peculiarities of present-day Pakistan.
I'm not sure that specifically religious history is the only explanation for why Christianity and Islam don't get along though. A lot of it is a lot more secular.
Both the Modern West (the civilization that Medieval Christendom has evolved into) and the Islamic world are very proud. They both see themselves as great and superior civilizations with a clear mission to expand their values and worldviews around the planet.
Islam was shaped by its amazing victories in its early years of expansion, when it defeated the Byzantines and seized most of their territory, and when they totally conquered the Persian Sassanids. Within a single lifetime, Islam emerged from the Arabian tribes and stretched from the Atlantic Ocean to the borders of China.
And that historically unprecedented success seemed to only be explainable by God's favor. It was proof to the Arabs that Mohammed was indeed a true divinely commissioned prophet.
And during the early Middle Ages, the Islamic world never slumped into as dark a "dark age" as most of Europe did. The Muslims had conquered two long-established empires and possessed an urban civilization with a money-economy from the very beginning. So the Muslims got in the habit of looking at the Christians as cultural and economic inferiors.
And once the Muslims were in that mindset, they retained it, even as the Christian High Middle Ages matched and then surpassed Islamic achievements. Even into early Modern times, the success of the Ottoman Turks, the Persian Safavids, and the Indian Moghuls, along with the rapid expansion of Islam in Indonesia, permitted the Muslims to concentrate on their own culture and achievements, while ignoring the increasingly annoying commercial developments and worldwide exploration coming out of the European fringe.
So they didn't pay much attention to the Scientific Revolution either. It was only the military superiority resulting from the Industrial Revolution that finally allowed the Europeans to overthrow most of the fading decadent and internally-divided Muslim powers and plant their own colonial rule atop them. The Muslims suddenly found themselves in what to them was the unnatural state of being the Europeans' colonial and cultural dhimmis. And that sudden failure came as a total existential shock to Islam.
Today's Islamic ferment is a reaction to that. Muslims are thrashing around for an explanation and a prescription that will right things and return them to the order that God intended. (Muslims on top.)
It's ironic that what Muslims typically think of as Islamic "modernism", is what we in the West label "Islamic fundamentalism". It's something like a belated Protestant Reformation, an attempt to sweep aside centuries of what many Muslims perceive as corrupt and decadent (and secular) traditions, so as to return to what they piously but probably unhistorically imagine to be the moral purity of the earliest Muslim community, loyal and obedient to God. And to that rightly-guided community's divinely-ordained worldly success.
I believe this reaction is historically disfunctional. It steers the new and more religiously militant Islam onto a head-on collision course with secular Global Modernity, both in the original Western version and the newer Confucian-inspired Asian version.