Via JewishAnswers.org:
Why is Rosh HaShanah before Yom Kippur if on Yom Kippur we repent for last year’s sins… shouldn’t we do it before the New Year starts?
Thank you for your insightful question. I would take this question even further and ask why should Rosh Hashana, the ‘Day of Judgment’ – judgment on the deeds of the past year – occur on the first day of the new year, rather than on the last day of the past year.
The answer lies in the way Judaism views the concept of Judgment. Judgment is much more than simply determining how much punishment or reward a person deserves in this world and the next. According to the Talmud, all of a person’s sustenance for the coming year is determined on Rosh Hashana – it is a judgment in the sense of taking account of all the factors in a person’s life and then ‘judging’ what he requires for the year, a complete assignment of one’s potential. In other words, it is the renewal of the year itself which triggers a period of stocktaking during which G-d reevaluates the present state of the world, and makes a judgment as to what is required for the coming year. Yom Kippur is simply part of this period of judgment and renewal and thus occurs after the new year has begun.