The concept of celebrating annual birthdays was not really a "thing" in early Islamic culture. It came into Muslim cultures through contact with cultures that did celebrate annual birthdays. Which leads to some interesting historical developments.

The Fatimids did celebrate Christmas, I believe as a state holiday, and the practice of celebrating Christmas by Muslims became widespread through areas that they controlled. Qur'an madrasahs would even get a full week off for Christmas! This led to a strong backlash by the religious class who were concerned about Muslims adopting non-Muslim celebrations. One theory (which I find very convincing) is that it was this very backlash that led to the now widespread practice of Muslims celebrating the Prophet Muhammad's birthday every year.

The oldest preserved Sunni book on the topic of celebrating the Prophet Muhammad's birthday is Kitab ad-durr al-munazzam fi i ‘l-mawlid al-mu’azzam which literally translates to "The Book of Edited Pearls Regarding the Birthday of the Exalted Prophet." The author actually lays it out pretty clearly, that kids grow up knowing all about the birthday of Jesus and John the Baptist but not about the birthday of the Prophet Muhammad. He also says that no amount of sermons and preaching can take the place of the joy children feel on holidays and festivals. Therefore, he writes,

“I have thought deeply as to how to distract the attention of the masses away from these innovated holidays and focus it on that which is permissible, that which does not cause one to observe sin....I have sought to draw their attention to the birth of the Prophet Muhammad." (translation mine)

As to why celebrating Christmas was seen as problematic by the Islamic scholarly class in the first place, it goes to the heart of mainstream Islam (Fatimids were a heterodox group theologically, a subsect of Shi'ism). If the Prophet Muhammad came with guidance for all time, then any religious ritual that a Muslim does must be based on his teachings. Religious festivities are included in that.

There are Muslims who celebrate Christmas. And some may even use the status of Jesus in Islam as a justification. But if you look at it historically, it's clear that celebration of these holidays has little to do with religious significance and more to do with social context (i.e., Muslims in areas with historically significant Christian population/influence may celebrate Christmas, Muslims in areas with historically significant Hindu population/influence may celebrate Diwali, etc).

References and more reading:

Marion Katz, The Birth of The Prophet Muhammad: Devotional Piety in Sunni Islam

N. G. J. Kaptien, Muhammad’s Birthday Festival