Luke's gospel tells us about how Mary and Joseph had to leave their home in Nazareth, Northern Palestine. They had to be registered in Bethlehem in accordance with Caesar Augustus' degree. Bethlehem was a little town near the capital city of Jerusalem in the south. Where in Mathew's gospel we are only informed that they lived in Bethlehem and Jesus was born there too.
"He went to register Mary, who was promised in marriage to him. She was pregnant, and while they were in Bethlehem, the time came for her to have her baby." (Luke 1:5-6 )
"Jesus was born in in the town of Bethlehem in Judea, during the time when Herod was king." ( Matthew 2:1 )
All throughout the birth of Jesus, in both accounts, you tend to find that there are many things that are different, rather few that are the same are similar. Another thing is, when you are told in Matthew's gospel that an angel had appeared to Joseph in a dream to say that he should marry Mary. The angel said, "Joseph, decendant of David, do not be afraid to take Mary as your wife. For it is by the Holy Spirit that she has concieved." ( Matthew 1:20-21 )
Then, it was in Luke's gospel were an angel, called Gabriel had appeared to Mary and not to Joseph. The angel came to her and said, "Peace be with you! The Lord is with you and has greatly blessed you!" Mary was rather troubled by the message from the angel. She had no idea what the words meant. The angel said to her, "Don't be afraid, Mary; God has been gracious with you. You will become pregnant and give birth to a son, you will name him Jesus." ( Luke 1: 28-32 )
You can see that both accounts are quite different in this case. Although, the similarity is that in both cases, the angel said that the child would be conceived by the Holy Spirit and would be called Jesus. Also, in Luke you're told that Mary went to visit her cousin Elizabeth, who had given birth to John. In Matthew you are noy told anything about Mary visiting her cousin.In Mathew's gospel you're not told that Mary and Joseph travelled to Bethlhem for the census, like I mentioned before, you find this in Luke's gospel.
The difference that I found rather interesting to mention, was that in Matthew's gospel were told that Jesus was born in a house, in Luke, he was born in a stable and laid in a manger. For me, the most difference between both accounts is that, Matthew tells us that wisemen had followed a star. This was after visiting Herod they found Jesus in Bethlehem and offered gifts, like gold, myrrh and frankincense. Gold which showed his Kingship, Frankincense showed that he would be like a priest and myrrh was to show that his death would be important. (Myrrh was put on to dead bodies to hide the smell of decay.) In Luke's gospel, an angel of the Lord had appeared to some shepards, and told them good news, that on the very same day the Savior was born, Christ the Lord. When the andels went away, all the angels hurried up and went off, and found Mary and Joseph and saw the baby lying in the manger. When the shepards saw him they told them that the angel said about the child.
After the birth of Jesus, "when the days of their purification according to the law of Moses were completed, they brought Him up to Jerusalem to present Him to the Lord." (Luke 2:22) According to the law of Moses after the birth of a male child a woman is unclean for seven days. On the eighth day the child is circumcised and the mother continues her purification for thirty-three additional days, or forty days total (Lev. 12:2-4). After that, from the forty-first day the child was to be presented at the Temple with a sin offering (Lev. 12:6-8).
The tradition of Bethlehem as the place of Jesus' birth has its source in all probability in the Old Testament conception of the Messiah as a decendant of David.
After Jesus' birth, in both accounts, it is told that Mary and Joseph returned to Nazareth with Jesus. Although in Matthew, Joseph was told by an angel in a dream that it would be safer to return to Israel to avoid Herod as he ordered the death of boys under the age of two, around all of Bethlehem. Jesus was in danger so that was the best choice for the time. A similarity between both gospels is that they describe Mary as a virgin, stating she had no
sexual intercourse with Joseph. Luke describes this on more detail.
So, to come to an end, Mathew's account was really the narrative which seems to be told from Joseph's point of view. Matthew wanted to establish the legitimcy of Jesus' claim to the messianic throne by taking Joseph's standpoint. Where Luke, apparently tells the nativty story from the stand point of Mary. He emphasizes her feelings and emotions as the mother of the Messiah. So, the differences between the birth stories in the gospels may have been for a variety of reasons, including the possibility that Jesus wasn't born in Bethlehem at all. However , the two birth accounts do inform us of God's intentions to minister to rich and poor, oppressed and mighty. And that Jesus, even as an infant, was Emmanuel God withn us. The two texts, however, cannot be harmonized.