Do Muslims Celebrate Christmas?

Some people ask why Muslims do not celebrate Christmas? Is that because they do not agree on the 25th of December as Jesus birthday? Or is that because they say Christians believe in Jesus as son of God, while Muslims do not believe in Trinity, as well as they believe that Islam is the religion that is more complete than Christianity and all other religions that came in the past; the same way that Christians believe that Christianity is a more complete religion than Judaism.