Luke 2

There are many differences between Matthew and Luke, not only in the story of Jesus's birth and childhood, but in the theological emphasis of those stories. I outlined those differences here: Early Events

Luke's gospel forms the foundation for most of our popular cultural ideas of the birth of Jesus. Luke is where we get the manger scene and the angels singing. The wise-men from Matthew are grafted into that, but not actually part of that version of the story. The truth is, the Bible presents two different stories for the birth and childhood of Jesus. There are a few good explanations that try to harmonize the stories, but that's really a study that you need to do for yourself if you're interested in it.

Luke presents Mary and Joseph as obedient and faithful Israelites. There's no mention of Herod, or any threat to Jesus's life. In fact Jesus is announced by angels and welcomed with open arms by shepherds. Where the magi returned home by a different road to avoid Herod, the shepherds returned to their fields glorifying and praising God.

Then Mary and Joseph took 8 day old Jesus into Jerusalem to the temple for Him to be circumsized. On their way into the temple they were met by two elderly prophets, one of them a widow, who praised God and spoke about Jesus. After that there's no mention of fleeing to Egypt, but we are told that they all returned to Nazareth.

Could Matthew and Luke both be telling the true story about the birth of Jesus? Yes.

Luke talks about the intimate details of Jesus's actual birth - the first days of His life. Matthew talks about the birth of Jesus in a way that could be summed up by the phrase "After the birth of Jesus" (Matt 2:1). Luke talks about details that sound like he got them possibly from Mary herself. Matthew talks about details that sound less intimate - the Magi speaking to Herod, Joseph having dreams (remember Mary was a widow by the time Matthew would have been writing), etc...

Mary and Joseph could have been visited by the shepherds at the manger, then took Jesus to Jerusalem to be circumcised, then returned to a relative's house in Bethlehem, then been visited by the Magi, then fled to Egypt, then returned to Nazareth, and both stories could be true - just focused on different details.

Then Luke adds in something that no other Gospel mentions - Jesus was found learning in the temple when He is 12. Not just sitting in silence as you would expect a child to do, but actually astounding the teachers there with His questions and understanding. And Luke keeps adding in that "Mary treasured all these things in her heart." I can imagine him talking to Mary and seeing her expressions as she told him each story, and being moved by how each event impacted her.

I try to read all the gospels together as a whole, but I can't deny that each one has it's own individual impact. Luke's gospel is one that strikes me as very personal and intimate - truly showing the human side of Jesus.