The calendar of saints commemorates holy ones mostly on the dates of their deaths, with two exceptions. The births of Mary, the Mother of Jesus and John the Baptist are both remembered, in addition to other feasts. From today’s Office of Readings, Augustine explains:

The Church observes the birth of John as in some way sacred; and you will not find any other of the great men of old whose birth we celebrate officially. We celebrate John’s, as we celebrate Christ’s. This point cannot be passed over in silence, and if I may not perhaps be able to explain it in the way that such an important matter deserves, it is still worth thinking about it a little more deeply and fruitfully than usual.

John is born of an old woman who is barren; Christ is born of a young woman who is a virgin. That John will be born is not believed, and his father is struck dumb; that Christ will be born is believed, and he is conceived by faith.

I have proposed some matters for inquiry, and listed in advance some things that need to be discussed. I have introduced these points even if we are not up to examining all the twists and turns of such a great mystery, either for lack of capacity or for lack of time. You will be taught much better by the one who speaks in you even when I am not here; the one about whom you think loving thoughts, the one whom you have taken into your hearts and whose temple you have become.
John, it seems, has been inserted as a kind of boundary between the two Testaments, the Old and the New. That he is somehow or other a boundary is something that the Lord himself indicates when he says, The Law and the prophets were until John. So he represents the old and heralds the new.

Because he represents the old, he is born of an elderly couple; because he represents the new, he is revealed as a prophet in his mother’s womb. You will remember that, before he was born, at Mary’s arrival he leapt in his mother’s womb. Already he had been marked out there, designated before he was born; it was already shown whose forerunner he would be, even before he saw him. These are divine matters, and exceed the measure of human frailty. Finally, he is born, he receives a name, and his father’s tongue is loosed.

Zachary is struck dumb and loses his voice, until John, the Lord’s forerunner, is born and releases his voice for him. What does Zachary’s silence mean, but that prophecy was obscure and, before the proclamation of Christ, somehow concealed and shut up? It is released and opened up by his arrival, it becomes clear when the one who was being prophesied is about to come. The releasing of Zachary’s voice at the birth of John has the same significance as the tearing of the veil of the Temple at the crucifixion of Christ. If John were meant to proclaim himself, he would not be opening Zachary’s mouth. The tongue is released because a voice is being born – for when John was already heralding the Lord, he was asked, Who are you and he replied I am the voice of one crying in the wilderness.

John is the voice, but the Lord in the beginning was the Word. John is a voice for a time, but Christ is the eternal Word from the beginning.

More from Catholic Culture, including this tidbit:

The Liturgy of the Hours for the Evening Prayer (Vespers) of the Birth of St. John the Baptist has traditionally included the Gregorian chant Ut Queant Laxis. Tradition has ascribed the hymn to a Paul Warnefried (Paul the Deacon, 730-799). While preparing to sing the Exsultet at the Holy Saturday vigil, he found himself hoarse, and so prayed to St. John the Baptist, since his father lost his voice before John was born. Paul’s voice was restored and he wrote this hymn in honor of the saint. True or not, what makes this song memorable is that the Benedictine monk used this hymn as a pivotal reference for our musical scale. See Catholic Encyclopedia’s entry Ut Queant Laxis, more information on the hymn from Catholic Culture, a Beginner’s Guide to Modal Harmony, and Gregorian Chant Notation.

There’s a generous snippet of the chant here, on the Amazon listing for the Chant II album.