Firstly, Merry Christmas. If I have readers of an Islamic persuasion, just think of today as the day we traditionally celebrate Saint Issa's birth. If you area Jew, please know that he wasn't born so his followers could lynch you-- you guys have really got to get over your fear of the goy. Remember: not everyone is out to put you to the sword!
No, Jesus/Issa/Yeshua wasn't born on 25 December; we just assigned it to 25 December because--
1. his original twelve disciples weren't that big on dates. It was enough that Issa was born, lived, worked, made wild promises about spiritual renewal and rising from the dead, died and kept those wild promises in their time.
2. the early church-- which had taken on the trappings of a Roman bureaucracy-- needed festivals and celebrations to happen simultaneously with the old pagan festival dates. All the better to infiltrate, co-opt and subvert (Hmmm, reads like America). With the introduction of Christmas, the sun god Mithra was shown the door, and his religion closed up shop.
I'm with the early Christians on the thorny subject of dates. It doesn't matter to me when exactly the Messiah was born. He came, cared about humanity, made wild promises and kept them. So.
There was a time when Christianity and Judaism were more tightly knit. Early Christianity in some quarters was treated as another sect of Judaism. There were--gasp!-- Christian rabbi preaching in the synagogues. Then the early church sunk its foot root in Rome and Constantinople. For some reason, the Christian Jews folded, died out. Christianity and Judaism have since taken separate roads.
Christendom does not celebrate the Savior's birth at the exact same time. The shape of our planet and its own rotation has seen to that. People live in different time zones, so the light from Christmas morning reaches people at different times, some (Asia) earlier than others (USA, Europe).