Every genealogy is a political statement. What I mean by that is that no name appears by chance. That said, look at some of the interesting differences between these two genealogies:
Matthew's lists four women, Tamar, Rahab, Ruth, and the wife of Uriah (a/k/a Bathsheba). Each of those stories proves to be an interesting subtext to the so-called Messianic line as it fits into Matthean portrayal of Jesus. Matthew's also begins with Abraham and has a distinctly numerological flavor, commenting on the importance of thrice fourteen successive generations.
Luke's lists no women, and as you noted does not agree with Matthew's. Additionally, Luke's genealogy goes to back to Adam "son of God" universalizing the Lukan Messiah.
I am sure more could be shown between these two, which apparently were using two sets of traditions that occasionally overlapped but often did not agree.
Euripides