Above: The familiar Nativity set
Below: Nativity scene based on Luke alone
The disparity between Matthew's and Luke's choices for witnesses to Jesus' birth (wise men and shepherds, respectively), tell us a lot about their focuses. Luke goes to great lengths to show that Jesus, although divine and conceived through the Holy Spirit, came to us humble and poor. Jesus was born in a stable and placed in a manger. There was no special star for Jesus, no powerful rulers coming to pay homage, and no expensive gifts. The witnesses to this miracle were mere shepherds. Since shepherding required long hours outside, but was not rocket science, often the youngest son of a family was burdened with the task. Notice how young the two shepherds in the painting are. They are just kids! Luke is interested in exalting Jesus by showing how he came to the world humbly and in poverty.
Matthew isn't as interested in that. He doesn't depict Christ as born into privilege, but he doesn't stress it like Luke. Jesus' birth, in Matthew, changes the order of the cosmos. Astrologers from the East see the star and visit Christ and pay homage with expensive gifts. Jesus, instead of being an anonymous birth in a stable, is seen as a political threat to King Herod, who orders a massacre of infants to quell any challenge to his power. Like Moses, Jesus escapes a decree to kill infants from a paranoid tyrant and is taken to Egypt (of all places).
There is another significant disparity in the two birth narratives, one that involves today's lectionary. In the Matthew lection (1.18-25), we are given the account of an angel appearing to Joseph in a dream, exhorting him not to break his engagement with Mary. The passage is not in Luke. In Luke's account, Joseph never has to be convinced to stay with Mary. Furthermore, in Luke the angel's commandment to name the child Jesus is given to Mary, not Joseph. Joseph is actually a relatively minor character in the birth narrative, as Luke devotes quite a bit of time to Mary.
Luke does something unique in devoting so much detail and thought to the women in the story. It is very much in tune with his emphasis on the marginalized, on those with a lower status. In fact, the person with the highest status in Luke's account -- Zechariah, the husband of Elizabeth and a priest -- is struck mute. He regains his voice after the birth of his son, John the Baptist. In yet another reversal of power, relatives wanted to name John Zechariah after his father, but Elizabeth insisted upon the name John. Women are given a special power and attention in Luke's narrative, and it is important to note that.
Matthew tends to focus on the males in the story: Joseph, the wise men, and Herod. Angels appear to Joseph in his dreams three separate times: 1) to foretell Jesus' birth 2) to warn Joseph to flee to Egypt and 3) to tell Joseph it was safe to return to Israel. The commonplace details present in Luke are absent in Matthew. However, the bigger issues absent in Luke are stressed in Matthew. The Gospel of Luke makes no mention of the massacre of the infants or the escape to Egypt.
The two accounts are not incompatible -- as the familiar Nativity scene above demonstrates, it is easy two cram the accounts together and add some harmless details (there is no mention of how many wise men came to visit Jesus). But when reconcile the two birth narratives, we should not forget the differences and details that make them unique. We have four Gospels for a reason: one perspective wasn't nearly enough.