The significance of Purim is so great that our Sages found allusions to other festivals contained within it. Thus:
Purim is like Pesach - on both we emerged from bondage to freedom.
Purim is like Shavuot - we accepted the Torah again on Purim.
Purim is like Rosh Hashanah - the books of the living and the dead were opened then.
Purim is like Yom Kippur - the generation of Purim then expiated their sins.
Purim is like Sukkot - just as Sukkot commemorates the protection accorded us by the Divine cloud of glory in the wilderness, likewise did many non-Jews enter under the protecting wings of the Shechinah during Purim.
Yom Kippur - A day like Purim;
So great is the merit of Purim, that Yom haKippurim, the holiest most solemn day of the year, is compared to it, (yom- a day; ki-Purim- like Purim). On Yom Kippur, Jews ascend to a level whereby they transcend the constraints of nature by denying themselves physical satisfactions and thus achieve atonement for the sins of the body. On Purim, this same level of holiness is achieved through eating and drinking, even man's physical pleasures become imbued with holiness.
When weighing the attainment of holiness achieved through denial and affliction against that achieved through engaging in pleasurable activities, the latter is considered superior, for it requires an infinitely greater measure of striving and effort. In this sense, then, Yom Kippur can be seen as being less than Purim, like Purim but not as great.
The commentaries point to another element that these two days share, albeit in inverted order. The fast of Yom Kippur is preceded by a mitzvah to eat and drink. Purim begins with a fast which is followed by a mitzvah to eat and drink.