I was told the following understanding of Rosh Hashanna and Yom Kippur yesterday. It was said by Rav Elya Ber Wacthfogel of South Fallsburg. I apologize for any mistakes and if you see any please let me know so I can make corrections.
There is an ancient question. Why does Rosh HaShanna precede Yom Kippur? We all know that for proper teshuva it is necessary to confess your sins (vidoi), have regret for sinning and we need to atone for those sins. Thus it would follow that if Yom Kippur preceded Rosh HaShanna that we could be properly judged as well as be prepared for the judgment. How can we be judged to be a tzadik on Rosh HaShanna when we have so much spiritual filth that we need to cleanse ourselves? A related question is the well known gemora (Kiddushin 49b) that if a wicked man betroths a woman on the condition that he is a tzadik - that there is a sofek that the marriage was valid. But the same queston occurs there. How could he be a tzadik without cleansing himself of all his sins?
The answer is that being a tzadik is not dependent on being cleansed of sin. It is an attitude that can be acquired in an instant. Becoming a tzadik and doing teshuva are two separate processes. A person who becomes a tzadik still needs to cleanse himself. However the cleaning up of the filth is a separate task and doesn't interfere with being a tzadik. Thus on Rosh HaShanna if we decide to be a tzadik then we are in fact a tzadik but the task of cleansing can be left over to the 10 Days of Repentance and Yom Kippur. The main issue is what we want to be - not how clear of sins we are. Therefore we first have the day of making a decision - which is the Day of Judgment - not the day of cleansing which is Yom Kippur.