In the December 2004 / January 2005 issue of Free Inquiry, Tom Flynn describes some of the basic differences:
The popular image of shepherds and wise men side by side before the cradle? Matthew says wise men. Luke says shepherds. Neither says both. The star in the East? Only in Matthew. "Hark, the herald angels sing" ... but only in Luke. Matthew never heard of them.
But then, only Matthew heard of Herod's slaughter of the innocents ... That's right, the indiscriminate killing of every male baby in Judea -- with one significant exception -- did not merit Luke's attention. On the other hand, no Roman historian chronicles this atrocity either, not even Flavius Josephus. Josephus reviled Herod and took care to lay at his feet every crime for which even a shred of evidence existed. Had Herod really slaughtered those innocents, it is almost unimaginable that Josephus would have failed to chronicle it.
Matthew says Joseph and Mary lived in Bethlehem, moving to Nazareth after their flight into Egypt ... But Luke says Joseph and Mary lived in Nazareth all along; Jesus was born in Bethlehem only because Joseph and Mary had traveled there to enroll in the census... Roman records mention no such census; in fact, Roman history records no census in which each man was required to return to the city where his ancestral line originated. That's not how the Romans did things.
It's not possible for both accounts of the nativity to be right. At least some important details of one or the other must be incorrect -- and if such important details are mistaken, the rest of that account must be treated skeptically as well. More serious, however, is the fact that so few people even recognize these problems. They combine elements of both stories into a single narrative as if they were all fully compatible. No one pays serious attention to just one account or the other. There's something very wrong when defenders of Christmas fail to take their own stories very seriously.