Do Luke and Matthew contradict each other or are they just different?

A:Yes, definitely. They do agree moderately well on the material both sourced from Mark's Gospel and the hypothetical 'Q' document, but material unique to Matthew disagrees strongly with material unique to Luke's Gospel.

We only have to look at the two nativity accounts which Raymond E. Brown (An Introduction to the New Testament) says are "massively different" and virtually irreconcilable. In Matthew's infancy narrative, Bethlehem was the home town of Mary and Joseph. There is no suggestion in Matthew of a census or of a journey from Nazareth to Bethlehem. The wise men found the young family in their house, not in a stable; they fled from Bethlehem to Egypt and after ther death of Herod began the journey back to Bethlehem but, being warned in a dream, turned aside and travelled to a city called Nazareth instead. In Luke's infancy narrative, Mary and Joseph lived in Nazareth in Galilee and travelled to Bethlehem, where Jesus was born, for a census controlled by the Syrian governor Quirinius; a few weeks after the birth of Jesus, the young family travelled to Jerusalem to present Jesus at the Temple, then returned peacefully to their home in Nazareth.

Another example is in the story of the empty tomb. Matthew chapter 28 says that just two women, Mary Magdalene and the other Mary, came to see the sepulchre. There was a great earthuake and an angel came down from heaven and rolled back the stone from the tomb and sat upon it. He told them that Jesus was risen, so they did not go into the tomb, but went quickly to tell the disciples. Luke chapter 24 says that Mary Magdalene, Mary mother of James and a number of other women brought spices to anoint Jesus. Finding the stone already rolled away, they went in and found the body of Jesus missing. Two men in shining gamrents told them that Jesus had risen.

In Matthew's Gospel, Jesus appeared only once to the disciples, in Galilee. Here, he told them to travle the world and spread the gospel. In Luke's Gospel, he met the disciples in Jerusalem on the evening of his resurrection and took them out on the road to Bethany, where he was taken bodily up to heaven. Once again, the disagreement here is total.