How do the infancy narratives of Luke and Matthew differ?

Significantly, in that Luke contains a lot of additional information not mentioned by Matthew. Matthew also records in his shorter account events not mentioned by Luke.

Matthew records the visit of the wise men, the flight into Egypt and the return to live in Nazareth.

Luke records the reason Mary gave birth in Bethlehem, (the decree by Caesar)and the visit to the shepherds by angels and their resultant visit to see Jesus. The circumcision of Jesus, together with the elderly Simeon and Anna are then followed by the return to Nazareth.