What are some differences between the Infancy stories in Matthew and Luke?

Matthew focuses less on the birth of Christ than Luke and moves quickly to the adult life of Christ. Luke is thought to have obtained much of his information from Jesus' mother Mary hence the extra detail. Matthew mentions Mary's conceiving by the power of the Holy Ghost and also includes the angel's appearance to Joseph. Luke includes details of the angel's appearance to Mary instead with the same information about the conception. Luke's extra information contains the visit of the shepherds and the presentation of the baby Jesus at the Temple.

Matthew doesn't give the story about Mary and the angel, but he goes into more detail about Joseph and the angel.

Luke's is more detailed. he talks about Mary and the angel Gabriel, and that they are in Nazarath.

Matthew doesnt talk about the visit to Elizabeth, but Luke does.