This much debatable question seems to have no definite answer. As nobody knows the exact date when Jesus Christ was born, there is no valid interpretation of celebrating Christ's birthday on a particular date. Christmas is a cheerful celebration of Christ’s birth but ironically, there's no written record of the occasion. The story of Christmas comes from the Bible which tells about the three wise men and the shepherds, who followed the Star of Bethlehem to reach the place where Jesus was born.
The practice of exchanging gifts started in memory of the gifts that the shepherds and the wise men brought for the Christ's child. Christmas, being the birthday of Jesus Christ, is celebrated with full gusto and fervor by all the Christians throughout the world. On the occasion, people exchange gifts and decorate their homes with holly, mistletoe, and Christmas trees. The term 'Christmas' is derived from the early English phrase 'Christes Masse', which means Christmas Mass.
For over 300 years, people celebrated Christ's birth on different days. While Clement of Alexandria preferred May 20 (c.150-c.215), Hippolytus (c.170-c.236) celebrated on January 2. Previously, there were people who celebrated the birth on April 18, 19, May 28, November 17, 20, and March 5 in the early days. In A.D. 336, when paganism was raging, Emperor Constantine desired to change the date and declared Christianity - the favored religion of Roman Empire.
In A.D 354, Bishop Liberia of Rome started celebrating the birth on December 25. This date was perhaps chosen because the Romans already observed this date as the Feast of Saturn, celebrating the pagan holiday "the birthday of the Sun." The Roman Emperor was convinced by the date as he wanted to make Christianity the preferred religion and this was the best way to celebrate Christ’s birth, taking over the pagan holiday.
The Roman Church purposely chose December 25 with the intention that Christ’s birth would replace the birthday of the Sun. Thus, the 25th was chosen not because the date was confirmed to be the actual birth of Christ, but to replace the pagan holiday. However, the Orthodox Christians still observe the festival on January 7. As per the records, no definitive research has been made to find out the actual date of birth of Jesus Christ.
Numerous people think it occurred during the fall, during the growing seasons at that time, however, there is no written record of the birth or death of Jesus Christ. It is just a speculation. Probably, the Emperor and Bishop might have schemed and decided on the particular date to celebrate Christ’s birth because of the pagan holiday.