Your question is a good one and often asked, so thanks for the opportunity to explain. According to the Mishnah, Rosh Hashanah is actually the beginning of the year, not only for the Jewish people, but also for all of mankind, as it coincides with and celebrates the creation of the first man and woman, Adam and Eve. The Mishnah relates that all of mankind is judged on that day, making it an international holiday, not only for the Jews. This is because on the day of man’s creation, all human souls are in a sense re-created, and are judged to determine if that re-creation should be for an entire year. This takes place on the first day of the Jewish month of Tishrei.
The month of Nissan, however, is the first Jewish month, as it marks the birth of the Jewish Nation. Passover, on the fifteenth of Nissan, is the celebration of that national birth which came through our redemption and freedom from Egypt. The Torah tells us to begin counting months from that month, as we have a unique role as Jews in the world, and are reminded of our mission through the way we count our months. To count from Rosh Hashanah would be to count as one of the “seventy nations” of mankind. To count from Nissan is to count as a unique nation, a light among the nations, which has been chosen to illuminate the world with the teachings of monotheism, morality, and Torah.
Rabbi Yerachmiel Fried