To any unbiased observer reading Matthew 1:18-2:23 and Luke 1:26-2:40, it's patently obvious that they're two completely different, incompatible stories of Jesus' birth. Here's a summary of the two versions:
Leaving aside the fact that the stories differ on almost every point, there are basically two direct contradictions. First, Matthew strongly implies that Mary and Joseph's hometown was Bethlehem, while Luke states that it was Nazareth. I went into this problem in more detail in a previous post. Second, Luke has them going directly from Bethlehem to Nazareth, while Matthew has Jesus' family fleeing from Bethlehem to Egypt. Errancy.org has more on this.
The birth narratives deal with quite a few alleged prophecies of Jesus. The first is Micah 5:2, which predicts that a savior would come from Bethlehem. Since Jesus was thought to have grown up in Nazareth, Matthew and Luke came up with different, conflicting ways to resolve this difficulty. But Micah 5:2 is referring to a tribe, not a town, and said savior was also supposed to defeat the Assyrians.
The birth narratives feature a virgin birth due to a misinterpretation of Isaiah 7:14, which appears to say someone will be born of a virgin. But the word translated "virgin" is more likely to mean "young woman," and the prophecy was already supposed to be fulfilled by Isaiah 8:3-4. The same verse also prophesies that "they shall call his name Immanuel," but there's no indication that Jesus was ever actually called by that name.
The gospel of Matthew is particularly big on attempting to fulfill prophecies. Matthew 2:23 says that Jesus' upbringing in Nazareth fulfills a prophecy saying "he shall be called a Nazarene," but no such prophecy appears anywhere in the Old Testament. Matthew 2:15 explains the flight to Egypt as a fulfillment of Hosea 11, which says, "Out of Egypt I called my Son." Yet looking at the original context, we instead see...
"When Israel was a child, I loved him, and out of Egypt I called my Son. As they called them, so they went from them; they sacrificed to the Baals, and burned incense to carved images."Not only is it not a prophecy of Jesus, but it's not a prophecy at all.
Culture and History
I want to quickly cover a few more points. First, some argue that Herod's massacre of the Bethlehem newborns didn't happen because it wasn't recorded by the meticulous historian Josephus, who recorded myriad other atrocities of Herod. Apologists reply that Bethlehem was so small that this would have been a minor event that Josephus may not have felt was significant enough to write about. I don't have enough information to conclude who's right one way or the other.
Then there's the Census of Quirinius, which is a massive problem for inerrantists. Here's a summary from Richard Carrier:
The Gospel of Luke claims (2:1-2) that Jesus was born during a census that we know from the historian Josephus took place after Herod the Great died, and after his successor, Archelaus, was deposed. But Matthew claims (2:1-3) that Jesus was born when Herod the Great was still alive--possibly two years before he died (2:7-16). Other elements of their stories also contradict each other. Since Josephus precisely dates the census to 6 A.D. and Herod's death to 4 B.C., and the sequence is indisputable, Luke and Matthew contradict each other.Finally, there's the Star of Bethlehem, which the Magi follow to Bethlehem. It's described as a real astronomical event—a star that rises in the east just as any star would—yet astronomers have not identified any event that matches its description, and it's unclear how a star could be situated directly above a particular building in a particular town. More importantly, the Magi's interpretation of the star is a form of astrology, which as Adam Lee points out is harshly condemned by the Bible. In fact, the very word "magos" literally means "astrologer." When the author of Matthew has a supposedly demonic power directing the Magi to worship Jesus, it's pretty clear that he's not on the same page as the rest of Christianity.
The birth narratives in Matthew and Luke are completely different and contain at least two major direct contradictions. At least four alleged prophecies either are not prophecies at all or are not fulfilled by Jesus. And there are multiple details that conflict with history or even Christianity itself. The birth narratives alone are more than enough to show that the Bible cannot be the inerrant word of God.