It’s impossible to miss the differences in Jesus’ ancestry as listed in Matthew and Luke:

• Matthew has three sets of 14 generations, with Jesus’ line traced back to David through his son Solomon. It’s a selective genealogy, written to make a theological point rather than write down a definitive family tree.
• Luke has many more ancestors listed, and Jesus’ heritage is traced back to David through his son Nathan.

The differences weren’t seen as a conflict for a few hundred years. It suggests that early Christians understood the genealogies and their meanings, and the conflict we now see is due to the loss of that understanding. I suspect that if there were a genuine conflict, it would have been edited out of existence very, very quickly—as soon as someone had the two gospels side by side—and we wouldn’t be seeing it now.

All we can do now is offer reasons for the differences between these two genealogies. Most are based around one being for Mary and the other for Joseph.

The most popular view at the moment is:

• Matthew has given Joseph’s genealogy by standard Jewish tradition of the day: it’s not meant to be a biological family tree, but a line of legal inheritance (that includes maternal grandfathers, for example, if a father dies early or has no sons to inherit).
• Luke has given Mary’s genealogy: Mary’s father had no sons, so when Joseph married Mary he became also ‘of’ Mary’s father to preserve the family line.

But there’s also the reverse view:

• Matthew has given Mary’s genealogy, on the basis of her having a father called Joseph as as well as a husband by the same name (we would need to find a really, really old original manuscript to support this, and we haven’t yet). Matthew also included women (Tamar, Rahab, Ruth, and Bathsheba) with interesting lives.
• Luke has given Joseph’s genealogy through (mainly) the male line because that’s what he says he’s done.

It’s actually easier to have two family trees than you think, even now. My own grandfather has both a biological father (who he never knew) and an adoptive father (whose name our family now has, although the adoption was never formal). I could publish completely different genealogies, and they would both be correct—but one is based on oral history (there’s not a scrap of documentary evidence for one of them being my great grandfather). In 200 years time, this will make any genealogical research on my family almost impossible.