A quick comparison of the issues I can recall at the moment, in order to share somewhere that tables aren’t conveniently available.
|Takes place before 4 BCE (while Herod the Great rules in Judea).||Takes place a decade later, in 6 CE (while Quirinius is governor of Syria).|
|Joseph and Mary appear to be living in Jerusalem.||Joseph and Mary live in Nazareth.|
|Jesus is born in a house (presumably, Joseph’s).||Joseph has to travel to Bethlehem due to a census. Jesus is born in a stable.|
|Jesus is worshipped by magi, apparently at least months later, since they had time to see the star rise at his birth and travel from the "East", and because when trying to kill Jesus, Herod has all children aged two and under killed, not just newborns.||Jesus is worshipped by shepherds.|
|(A month after Jesus is born, the magi can’t even have arrived yet!)||A month after Jesus is born (after the rites of purification), the parents return to Nazareth.|
|Afer the magi leave, the family flee to Egypt to avoid Herod [the Great].||(Nothing.)|
|After Herod dies, the family return to Israel, but rather than return to Bethlehem, they move to Galilee, because Herod’s son Herod Archelaus rules in Judea.||(Nothing.)|
There are some additional problems.
- Matthew actually begins with Jesus’s alleged genealogy. This is largely drawn from the Old Testament, but because Matthew has a theological notion that there’s something special about blocks of 14 generations between important people, he actually leaves out a number of people in the Old Testament genealogies.
- Supposedly, the magi find the house when the star stops over it, but…well, obviously, a star cannot stop over a specific house; even if it’s something like a comet or magical meteorite, you can’t possibly pinpoint a specific house by its location.
- Even though historians like Josephus wrote very critically about Herod and catalogued a number of his bloody deeds, there’s no mention of an indiscriminate slaughter of all the toddlers in Bethlehem. This was probably invented as part of Matthew’s “New Moses” narrative.
- Galilee is strange place to flee to, because Herod’s other son, Herod Antipas, ruled Galilee: but the gospel is explicit that this is why they move to Galilee rather than return to Judea.
- It is a bizarre census that can’t have happened as described: there was no empire-wide census, and for taxation purposes it makes no sense to have people return to the putative birthplace of their ancestors a thousand years prior and register there, rather than where they are being taxed, which is where the information is wanted! That’s even if it were logistically possible, which of course it wasn’t: how would everyone know their lineage going back that far? how would everyone be uprooted to traipse off to ancestral home towns? In short, it’s a census that could not have been made to happen, and would have been a terrible idea even if it could.