The genealogies of Matthew and Luke differ in a number of ways.  As Peter J. Williams notes: ‘Matthew’s genealogy runs from Abraham down to  Jesus in three groups of 14 generations. Luke’s genealogy goes from Jesus all the way back up to Adam, and indeed God before him.  While the two genealogies are similar between Abraham and David, they diverge dramatically between David and the exile, meeting for Shealtiel and Zerubabel, before diverging again and only meeting with Jesus’s legal father, Joseph. Consequently, Joseph is presented as having two different fathers: Jacob in Matthew and Eli in Luke.’

Various explanations for the divergences have been offered.

(1) One (probably Matthew) records the ‘official’ genealogy through Joseph, the other the ‘actual’ genealogy through Mary.  This is the view of Howe and Geisler (When Critics Ask).  Matthew addresses the interests of his mainly Jewish readers, who needed to know that Jesus was the promised Messiah.  Luke addresses those of his mainly Greek readers, who would be more interested in Jesus as the Perfect Man.  However, there is nothing in the text that would suggest this.

(2) One (probably Matthew) spiritualizes the genealogy rather than following it literally.

(3) The lines of descent cross but are different because one list includes several adoptive lines through levirate marriages (Deut 25:5-10) (NT Background Commentary).

F.F. Bruce (Answers to Questions, p40f) inclines to the view that Matthew’s genealogy gives the line of succession to the throne of David (which did not always correspond to the line of descent from father to son, whereas Luke gives the line of Joseph’s descent from David by another branch of the family.  Machen is quoted: ‘There is nothing at all inherently improbable in such a solution.  When a kingly line becomes extinct, the living member of a collateral line inherits the throne.  So it may well have been in the present case.’

Peter J. Williams stresses that ancient genealogies do not necessarily conform to modern ideas of such things: ‘To us, genealogies record faithfully and accurately our lineage, step-by-step, through the generations, without missing any out.’

Williams continues: ‘The two gospel genealogies make different points. Matthew traces from Abraham through the royal line to Jesus, and strikingly mentions four women who were either non-Israelite (Tamar, Rahab, Ruth) or at least had foreign connections (Uriah’s wife). Among other things this prepares us for the end of the book, which shows the gospel going to all nations.  Luke’s genealogy connects Jesus with the first man and helps us to think of the contrasts between Jesus and Adam (and all other humans generally). It’s a perfect prelude to the temptation narrative in Luke 4:1–13 in which Jesus refuses food in the barren desert in contrast to Adam who took the forbidden fruit in a garden full of other fruit.’