There is a famous statement of the ARI'ZL (Rabbi Yitzchak Luria, a sage who lived in the 1500's and studied the mystical aspects of the torah) which states that the hebrew words “Yom Kippurim יום כיפורים contain an allusion to Purim. If we break down the word Kippurim כפורים into a shorter word that is introduced by a prefix, i.e.-Ki-purim, we get a word that means Ki-purim כ-פורים “like Purim”. The ARI'ZL explains that this allusion teaches us that Yom Kippur is only like the holiday of Purim but not as great. This statement of the ARI'ZL requires clarification. How can it be that the holiday when Jews from all walks of life come to shul and pray a whole day to Hashem while fasting the entire time, is subordinate to the holiday of Purim? On Purim we spend most of our day eating , drinking , and making merry!
Perhaps the distinction between Purim and Yom Kippur can be understood as relating to Hashem's differing conduct on these two holidays.
The Ran, in his commentary on Masechet Rosh Ha'shana asks why Hashem chose the tenth day of the month of Tishrei to be the day of atonement? The Ran answers that the tenth day of Tishrei was the day that Hashem forgave the Jews for the terrible sin of worshiping the golden calf. After 80 days of Moshe pleading and begging with Hashem until Hashem finally agreed on the tenth day of Tishrei to forgive the Jews. Since this sin was so severe (Hashem had wanted to wipe out the entire Jewish nation as a punishment for it), the fact that Hashem agreed to forgive the Jews was a tremendous manifestation of divine compassion. This caused, says the Ran, the tenth day of Tishrei to be imbued with the power of forgiveness. Therefore Hashem chose this day to be the day of forgiveness and atonement, Yom Kippur, for all generations.
On the other hand, the holiday of Purim commemorates an even greater level of compassion on the part of Hashem. The Peleh Yoetz (ערך פורים) points out a phenomenal thing about how the story of Purim unfolded. He cites the gemarah in Masechet Megillah (12a) which says that as a punishment for partaking in the unkosher feast of Achashverash the Jews were sentenced in heaven to extermination at the hands of Haman. Nevertheless, says the Peleh Yoetz, at the very same time that the Jews were publicly sinning and arousing G-d's anger, what was Hashem doing? Hashem was setting in motion the process that eventually led to the Jew's salvation. Hashem “arranged” for Achashverash to get drunk and make the ridiculous command that Queen Vashty his wife appear unclothed in front of his officers. This, in turn, led to Queen Vashty's execution and set the stage for Esther to fill her place. The incredible thing about this was that Hashem was showing compassion at the very moment that they were committing a sin of huge magnitude. This is a much greater display of mercy than what happened by the sin of the golden calf. The latter was mercy that came about after the sin was over, and after a long process of Moshe asking for forgiveness. In contrast, the mercy that Hashem showed on Purim was mercy during the time of the sin itself. This is a much higher level of divine mercy.
In light of this observation we can now suggest a novel explanation of the ARI'ZL's statement that Purim is greater than Yom Kippur. The ARI'ZL meant that the amount of divine mercy available on Purim is greater than the amount of divine mercy available on Yom Kippur.