The angel Gabriel tells us that Zechariah did not believe while Elizabeth tells us that Mary believed. While the words of Zechariah and those of Mary are very similar, they express a radically different attitude. In Mary, by contrast with him, we find a receptivity that makes the birth of the Word possible.
The one to whom Mary gives birth is the Holy One, the Son of God. The sign given to Ahaz, of which Isaiah speaks, was the birth of Hezekiah, a good and righteous king, faithful in his service of the Lord, in this way a contrast to his father, Ahaz. Mary also gives birth to a good and righteous king, Him whom the liturgy describes as the 'fountain of all holiness' (Eucharistic Prayer II) and the source of 'all life, all holiness' (Eucharistic Prayer III).
So God visits His people and reveals His glory anew. Notice also that Zechariah is in the Temple while Mary is at home in Nazareth. This new visitation and revelation does not happen in what seems like the obvious place, the Temple, the place of the presence of God's glory. It happens where there is a receptive heart. We need to beware of any proprietorial attitude towards God and His glory. Perhaps this is the crucial lesson to be learned from Zechariah's mistake: how do we leave space for God to do a new thing, God who, precisely because he is ever-faithful, is ever-creative? The Word can only come to birth where there is a believing heart. That is the space in which God can do a new thing.