The versions of the nativity given by Matthew and Luke are strikingly different in their details, and we can gain insight from looking at each separately, rather than collapsing them together as we so often do. While Matthew largely gives us the story of Joseph, Luke largely gives us the story of Mary, and if Matthew stresses the regal and prophetic elements of Jesus' birth, Luke stresses the humble and human elements.
Luke starts with a parallel story: the births of John the Baptist and Jesus. John is Jesus' forerunner from his very conception, and each element in John's story is recapitulated and heightened in Jesus' story. John's birth is foretold by an angel to Zechariah, and the angel tells him of the importance of the child's life and ministry. Six months later, Jesus' birth is foretold by the angel Gabriel to Mary, a virgin in the rather low-class town of Nazareth. The angel tells her that she will have a child who will "be called the Son of the Most High" and will be given "the throne of his father David," and his "kingdom will never end" (1:32-33). We have here a stark contrast between what is now and what is prophesied to be in the future.
Mary visits her cousin Elizabeth, the mother of John the Baptist, who acknowledges the greater importance of Mary's child. Mary, for her part, gives voice to what we now call the Magnificat: essentially, a psalm of praise to God and a recognition of the great things God will do through her son, in large measure conceived as overturning social and economic inequalities. She voices the longing in the hearts of a downtrodden people, looked down upon by their countrymen in Judea and treated as captives by the Romans.
Luke narrates the birth of John the Baptist, and then turns to the circumstances of Jesus' birth. While Matthew relates the conflict with Herod after the child's birth, Luke relates the unwitting involvement of the Roman Emperor himself in the circumstances of the birth. A census is taken, in which all are required to return to their home towns to register; Joseph, being descended from King David, returns to David's home town of Bethlehem, and Mary comes with him. It is here in Luke that we find that Jesus was born, apparently, in a stable, because a feeding trough was improvised as a crib. Although from a royal line, Joseph and Mary are reduced to giving birth in the lowest possible circumstances.
In Luke, Jesus' birth is witnessed, not by foreign dignitaries, but by shepherds, who are socially outcast and ritually unclean by virtue of their work, and yet reminiscent of David. The birth is announced to them by angels, the shepherds come to Bethlehem, and there is no doubt that they in fact were there on the night of Jesus' birth, because they saw him in the feeding trough. Unlike the visit from the Magi, these witnesses draw no attention from the earthly authorities, although the shepherds do spread the word of what they have seen and heard.
Joseph and Mary fulfill the Law of Moses by having Jesus dedicated at the Temple and by offering the sacrifice appropriate to the poor. His destiny as savior of his people is attested to by Simeon and Anna--not the priests or temple officials, but elderly righteous people who were there at that time. Luke doesn't relate the flight to Egypt, simply collapsing the narrative down into "they returned to Galilee to their own town of Nazareth" (2:39).
So by contrast to Matthew's narrative of Jesus' birth breaking into and threatening the earthly powers, Luke relates the humble, even humiliating circumstances of Jesus' birth. An underclass of people, waiting at the brink of hopelessness for a savior. The news of his birth, far from having to remain a secret, is spread indiscriminately--because the class of people who know are irrelevant to the powers that be. But there is hope: echoes of David the shepherd-king, raised from obscurity to power, a man after God's own heart.