Problem: Matthew and Luke's nativities are totally different

An angel appears to Joseph to reassure him, and so he marries Mary.
Jesus is born in Bethlehem.
Perhaps two years later (or perhaps not), wise men see his star. They come and inform Herod.
The wise men - bringing gifts - find Jesus in Bethlehem.
Warned in a dream, Joseph and family flee from Bethlehem to Egypt.
Herod commences the massacre of the infants.
Herod dies. Informed in a dream of Herod's death, Joseph takes the family back.
But he is afraid to go to Judea, and so makes his home in Nazareth, Galilee.

A census requires Joseph and Mary to go from their home in Nazareth to Bethlehem.
Jesus is born in Bethlehem.
There is "no room in the inn"; Mary places Jesus in a manger.
Nearby shepherds are told of these events by angels.
The shepherds visit the family.
After about a month or so, Jesus is taken to temple in Jerusalem.
There, Simeon and Anna praise Jesus.
Soon after, Joseph and Mary return to their home in Nazareth.

The first thing to say is that these are obviously different stories. There is just no overlap between them except for the fact that they feature the birth of Jesus in Bethlehem, and end up in Nazareth. Even if the two accounts had no explicit contradictions, it's decidedly odd that almost nothing that occurs in Matthew is mentioned in Luke, and vice versa.

Where was the home of Joseph and Mary?

Before anything happened, where was the home of Joseph and Mary? Luke 2:4 and 2:39 are perfectly clear: it was in Nazareth, Galilee. On the other hand, Matthew doesn't explicitly say. But he seems to place it in Bethlehem, Judea. Firstly, he never describes any journey to Bethlehem (presumably because Joseph and Mary were there already). Secondly, when Herod dies and Joseph is taking his family back home, it seems he meant to go to Judea (where Bethlehem was), but was afraid to do so. So he went to Galilee instead, and made a new home in Nazareth. (See Matthew 2:22-23.)

This is not quite conclusive. Just as a matter of geography, one would generally want to pass through Judea in order to get from Egypt to Nazareth, and so when Matthew tells us they avoided Judea, he might just be discussing the route they took on their journey, rather than their intended destination. If you consider Matthew in isolation from Luke though, you are definitely left with the impression that Joseph had intended to go home to Bethlehem.

How long were they away from home or in Bethlehem?

A more difficult problem is this. Matthew's gospel tells us that the family had to flee from Bethlehem and hide in Egypt for a while. In Luke, however, once the month-old Jesus had been presented at the temple, and all legal requirements had been fulfilled (see Leviticus 12), the family simply returned to Nazareth. This is Luke 2:39:

And when they had performed everything according to the Law of the Lord, they returned into Galilee, to their own town of Nazareth. (ESV)

This really can't be reconciled with Matthew, who says they did nothing of the sort, but rather fled to Egypt. Also, Matthew 2:16 seems to imply that the family was in Bethlehem for about two years before they had to make their escape, whereas Luke has them stay in Bethlehem only a short while.

The Tektonics website tries to solve both problems at once, by suggesting that the family went to live in Nazareth (as Luke says), but just happened to be visiting Bethlehem for unrelated reasons when the wise men went there looking for them. Quite a coincidence! But, there's nothing in Matthew to indicate that Joseph and Mary left Bethlehem prior to the wise men finding them (which makes a lot of sense if Matthew thought their original home was there).

Looking Unto Jesus suggests that the family did indeed go to Nazareth after about forty days, and then fled from Nazareth (not Bethlehem) to Egypt some time later. Again, it should be stressed that there's nothing in Matthew to indicate this: Nazareth is not even mentioned till later. In any case, if Herod was looking for Jesus in Bethlehem, there would be little reason to flee from Nazareth, which is quite far from Bethlehem - at least 60 miles or so.

So what's going on?

What was actually going on in the minds of the authors of Luke and Matthew? It seems that everyone knew that Jesus came from Nazareth, a town not mentioned in the Old Testament. However, an important Old Testament prophecy (Micah 5:2) placed the birth of a great King of the Jews in Bethlehem. It seems that both Luke and Matthew wanted to resolve this little problem, but did so in very different ways.