"The genealogies of Jesus in Matthew and Luke appear to be different.  Why?"

You will find Jesus' human family tree listed in Matthew 1.2-16 and Luke 3.23-37.  Matthew starts the genealogy with Abraham, while Luke lists the names in reverse order, starting with Jesus going all the way back to Adam.  In conformity to ancient practice, both lists of names occasionally skip generations, listing the most prominent names in the family tree.  But the most glaring discrepancy is that the two lists of names between David and Joseph are completely different.  Matthew's list includes every king of Israel in David's dynasty, while in Luke's list David is only king named.  It appears that Jesus can track His human family line back to David through two branches.

There are several reasonable explanations for the difference.  First, we do not know any details of Joseph's parentage. He may have had both a biological father and an adopted father, either of whom he could legally claim as his father.  But this suggestion is merely a guess and does not answer a more important question about Jesus' family tree.

Since Jesus was born of a virgin mother, we understand that Joseph was not Jesus' true father.  Jesus' biological descendancy from David must come through Mary.  When we compare the Christmas stories in Matthew and in Luke, we see that Matthew tells Joseph's story, while Luke tells Mary's story.  So it is reasonable to assume that Matthew presents Joseph's lineage, fulfilling Old Testament prophecy that Jesus was legal heir to David's throne, while Luke identifies Jesus' biological descendancy from David through Mary's branch on the family tree.  Since Joseph must be named as Jesus' legal father, Joseph's inclusion in this branch is by way of marriage, making him Heli's son-in-law.

The puzzle that the differing genealogies presents reminds us to consider Jesus' two natures.  He is both fully God and fully human, the only One who could be our substitute on the cross and become our Savior.