The key point is that Luke is writing to a Roman official and Matthew is writing to Christians who were formerly Jews. In other words, Luke is writing to the oppressor and Matthew is writing to the oppressed. Luke has carefully left out those things that would upset the Roman official, Theophilus, or any other Roman official that Theophilus might show Luke's gospel. Matthew has similarly left out those things that would upset Jewish Christians.
As a Catholic I know by faith that there was no dishonesty, just good prudent editing. This kind of prudence is not sinful, in fact it is at least recommended if not commanded by the Bible. Paul when instructing Christians on how to speak to those outside the faith said, "...try to fit your answers to the needs of each one." Colossians 4:6. Jesus told us to be, "... be cunning as serpents and yet as harmless as doves."; Matthew 10:16. So Christians should not be shocked or scandalized if the Holy Evangelists exercised an honest, godly, prudence in their editing.
Matthew does not mention Nazareth at first. He briefly deals with the virginal conception and birth of Jesus and then rushes on to the wise men, Herod, the slaughter of the Holy Innocents, and the flight to Egypt.
In Luke, the wise men, Herod, the slaughter of the Holy Innocents and the flight to Egypt are all skipped. Jesus is born in Bethlehem, is visited by the shepherds, is circumcised, goes Jerusalem for Presentation at the Temple, and then the Holy Family goes back to Nazareth.
What really happened? Jesus was miraculously conceived by the Holy Spirit in Nazareth. Matthew leaves this out because it would weaken the Christian claims that Jesus was the Messiah who had to be from Bethlehem, the city of David.
Luke emphasizes the same point Matthew leaves out because Jews claiming to be the Messiah were always leading revolts against Rome. So writing to a Roman official Luke plays the Messiah aspect of the Christian faith down without denying it.
Jesus was born in Bethlehem, fulfilling the prophecy that he would come from Bethlehem. Both Matthew and Luke mention this. Luke might have wanted to leave it out, but it was simply too central to the story.
Luke mentions the shepherds' visit to the Holy Family, Matthew does not. As shepherds were poor marginal members of society this would make Jesus, and therefore the Christians, less threatening to the Romans.
Luke says that Jesus was circumcised on the eighth day, and about a month later was presented at the temple, and returns to Nazareth. Matthew leaves out both the circumcision, the Presentation, and the return to Nazareth. At the Presentation the Holy Family provides two turtledoves or two young pigeons for the sacrifice. This was the sacrifice of the poor. If they had just received gifts of gold, frankincense, and myrrh then they would have provided a lamb. This is one of the clues that provides evidence that the wise men showed up later.
As devout Jews the Holy Family made at least yearly pilgrimages to Jerusalem, which is a few miles from Bethlehem. Joseph had relatives in Bethlehem and there might be carpentry that he could do there. So it would be only natural that the family would visit Joseph's relatives in Bethlehem, and perhaps Joseph would pick up some work, which might make the stay longer. It was during such a visit that the Wise Men showed up.
As Herod was an official appointed by Rome, Luke naturally left out the slaughter of the Holy Innocents. Starting your story of the life of Jesus by mentioning what was at least indirectly a Roman atrocity would have be very imprudent. Luke's decision to spare the feelings of the Roman official on the slaughter of the Holy Innocents is probably the key reason that the stories are so different.
Matthew on the other hand was writing to Christians who had formerly been Jews. They would have been favorably impressed by story of the Wise Men. The Romans on the other hand might suspect that the Christians were making alliances with powerful people beyond the boarders of the empire. They might have reacted negatively to the story of the Wise Men.
The Holy Family fled to Egypt and eventually returned to Nazareth.
A consistent story can be constructed from the two gospels if we recognize that both writers were writing to two very different audiences. They were not testifying before a modern court, they were writing the truth, but they had not promised to tell the whole truth. They picked the parts of the true story that would help them win over their very different audiences.
Some modern scholars have tried to make Luke into a 1st century feminist because he writes so much about Mary. More realistically Luke concentrated on women, and men past military age, like the father of John the Baptist, because that was less threatening to the Romans.
Proper Biblical exegesis requires that we read the Bible in the terms of the political issues of the time it was written, not in terms of the politics of our time. Biblical scholars insist on this principle and then violate it wholesale.
It has often been noted that the genealogies of Matthew and Luke do not match. The exact same analysis that I have applied here, may help solve that mystery also. Luke presents a genealogy that de-emphasizes Jesus as the King of Israel, Matthew does the opposite. Check it out on my web page on the genealogy of Jesus.
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