Had the priests won in their debate with the sages, today's Jewish calendar would look very different.
According to the priestly solar calendar, the first day of the year fell in spring, on the vernal equinox, not in the fall where the rabbis placed Rosh Hashana. In the priestly calendar, the Day of Atonement always fell on a Friday, the Festival of Unleavened Bread (Hag Hamatzot - a home holiday which followed by a day the Pessah Temple holiday) - and Succot always began on Wednesdays, Shavuot always on a Sunday and no festival could ever fall on Shabbat.
With a large measure of spite, the sages turned all of this upside down with their lunar calendar so that none of the holidays would fall on the days prescribed by the priests. Any holiday could fall on Shabbat. Yom Kippur can never fall on Sunday, Wednesday or Friday, the priests' chosen day. Hag Hamatzot can never begin on Friday, Monday or Wednesday, the priests' chosen day. Rosh Hashana can never fall on Sunday, Friday or Wednesday, the day it begins in the priestly calendar.