Why aren't Christmas and New Year's Day the same date?
In one sense they are the same date!
The short answer is that they are simply not taking the same place within the civil calendar.
Yet on the other side of the coin, they may seem to be the same date, although their not.
This question is going to be answered in a little convoluted and complicated manner.
It is true that liturgically speaking the Feast of the Circumcision of Christ is celebrated on January 1st, while the Feast of the Nativity is celebrated on December 25th.
In the ancient days of Christendom, major feasts were celebrated for eight days. Christmas in days gone by was celebrated for eight days in a row, called liturgically an octave. Thus the Feast of Christmas columnated with the Solemnity of the Circumcision of Our Lord Jesus Christ. Thus in a liturgical sense of traditional minded Catholics, Christmas is one continuous liturgical feast that ends on January 1st.
The Feast of the Circumcision is a natural way to conclude the Christmas feast day, as circumcision was performed on the "eighth day" after birth, the number 8 became associated also with baptism, and baptismal fonts have from an early date often been constructed in an octagonal design.
Thus the Brit Milah of Jesus would be a fantastic way to end the Octave of Christmas. Satan hates this day for on this day Christ received the sacred name of Jesus. In the Catholic Church, January 1st remains a holy day of obligation.
Thus we can conclude the Pope Gregory XIII believed the Jesus was the Christ at the very moment of his conception in the womb of the Virgin Mary. Moreover, Christmas and the Circumcision of Christ was seen as one continuous feast!
In 1582, Pope Gregory XIII (Gregorian Calendar) corrected the Julian Calendar, the first day of the year varied from country to country.
For example, England celebrated New Years Day on March 25th. Now in England, the 25th of March fell on the Feast of the Announciation. The faithful of the Middle Ages were more religiously mind than our modern society is now. This can be seen in the fact that in som pious liturgical books the Annunciation and Good Friday. This had no effect on the date of the Easter celebration which was liturgical celebrated anytime between Marsh 22nd to April 25th, depending on when the first full moon of spring occurred.
It should equally be noted tha the Romans celebrated their New Years Day on January 1st. In an other view point, Gregory XIII was restoring the first day of the civil calendar to it’s original place!