(a) Apologetics information on harmonizing the timing: While the two accounts from Matthew and Luke can be mostly made to harmonize into a single timeline, the dates are confused by about ten years if you accept the Josephus date for the census of Quirinius. There is reason to believe that the date of the census given by Josephus is incorrect (source). Furthermore, the word “first” implies that the one mentioned by Josephus is a “second” or subsequent one. The Greek of Luke also doesn’t give Quirinius a specific title, and it may be that he conducted the census before becoming Prefect or that the one mentioned by Josephus is as part of a second term (pdf, source). Pulling this information together, here are some resources about establishing a specific date.
(PDF versions saved for posterity)
(b) While harmonizing is important from a historical standpoint, from a literary perspective, the contrasts are interesting in examining the thrust of each author. Matthew presents Jesus as being part of a well-to-do family with resources to travel and a home, is visited by Magi, traveling freely under the direction of angels. Matthew makes the bad guys the Jewish betrayer Herods. The flight to Egypt sets up a contrast between Jesus and Moses, as well as the “to all nations” conclusion. Luke presents Jesus as being of lesser means, with no space in the inn and visited by shepherds, traveling under the compulsion of the decree of the Romans. Luke makes the bad guys the external Roman conquerors. The temple encounters show the prophetic spirit and therapeutic emphasis of Luke.
(c) We are currently in a series on Luke, so with the Lucan distinctives in mind, what can we understand that this gospel wants to say to us or ask of us.
Jesus is with the lowly and humble of heart. Am I so?
The appearance of Jesus brought great joy to people of every station. Do I have joy at the appearance of the Lord in my life still?
The people he appeared to were all eagerly awaiting his coming. Am I waiting with hunger and thirst?
In all cases, the people went out glorifying God and proclaiming that redemption had come to Israel. Am I faithful in proclaiming the message?