In this U.S. Catholic interview, scripture scholar Sr. Laurie Brink, OP provides some very sensible, scholarly, and faith-based pastoral answers to questions you may have wondered about the Gospel accounts of Christmas–but were afraid to ask. For instance:

Why do the stories differ in these significant details?
One way to account for differences is that the authors are talking to different people with different interests, and from the ancient Church Fathers on, most people have presumed that there were different communities.
Matthew’s community is probably Jewish-Christian, well versed in the Septuagint, the Greek version of the Hebrew scriptures. That’s why he cites the prophet Isaiah. Those make sense only if you know what he’s referring to.
Matthew’s audience is probably deeply rooted in their Jewish past, and he is helping them connect that heritage with Jesus. He parallels Jesus with Moses. That typology only works if you’re familiar with the Moses story.
Luke’s community is probably more Gentile. Luke certainly has references to the Hebrew scriptures, but they’re more subtle. His audience knows the scriptures the way Catholics know the Bible: We know it generally.