Why did John the Baptist exist? Did he have any purpose besides Baptizing Jesus? I realize this question is somewhat vacuous, since one could ask:

"Why do the OT prophets exist? Why wasn't the story just: Eve + Adam eat from the tree of knowledge of good + evil. Then, God, immediately sends Christ down, who dies for humanity's sins, QED."

Nevertheless, what is the purpose of John the Baptist's existence / ministry, given that it occured in the same time period as Christ?

There is a very significant even of John Baptizing Jesus:

Matthew 3:13-17


  13 Then Jesus came from Galilee to the Jordan to be baptized by John. 14 But John tried to deter him, saying, “I need to be baptized by you, and do you come to me?”
  15 Jesus replied, “Let it be so now; it is proper for us to do this to fulfill all righteousness.” Then John consented.
  16 As soon as Jesus was baptized, he went up out of the water. At that moment heaven was opened, and he saw the Spirit of God descending like a dove and alighting on him. 17 And a voice from heaven said, “This is my Son, whom I love; with him I am well pleased.”


Besides that, what is John the Baptist's contribution/significance/utility to Christianity?
I can think of a few reasons but there are probably much more.  

First, as Jesus was not ordained by the existing authorities and seemed on the surface to take things in a different direction from the Old Testament, a forerunner which was prophesied to herald his coming, not only fulfills prophecies of Elijah's return but creates a smooth transition from the Old to the New Covenant where the prophets directly lead up to the entire end of all their prophecies in Christ. What I mean is John was like a typical prophet denouncing sin and calling for repentance with a kind of rugged Old Testament asceticism and harshness, somewhat different from the meek and mild Jesus that pronounced peace and grace.  When the Jews questioned Jesus on how he derived his authority to preach and to heal (since he was not recognized by the 'authorities' of tradition) he asked them what they thought of John the Baptists (Mark 11:29). The question was intending to show them that they were so cowardly (not admitting that they did not believe in John the Baptists among the crowd) that they were not worth responding to, but still the connection to the Baptists is made regarding recognition of an 'authority', besides the direct testimony of God the Father of course.

Second, that a great prophet such as John was, in his humbly recognizing Jesus as so much greater, properly introduced the Messiah. (Matthew 3:11)

Third, his baptism of the Jews by John was a kind also used to convert Gentiles to Judaism, so like Elijah he was making a great call to repentance communicating to everyone, even those who thought they were holy, must prepare for full repentance and be prepared to receive the Messiah.  This mission was like Elijah. 

Fourth, The idea that Elijah would come just before the Messiah (literally Elijah, not someone like him) was completely expected by rabbinic tradition, so the fulfilled prophecy was no small matter, even though, as many of the other fulfilled prophecies, we're not fulfilled in ways expected, as a whole they became a strong testimony on Christ's authenticity.