Keep in mind that almost everything I know about this case comes from the musical Hamilton, which is more than likely to include historical inaccuracy. That being said, if the account in the musical number "We Know" is accurate, Thomas Jefferson, Aaron Burr and James Madison reassured Hamilton that they wouldn't publicly out him for his affair with Maria Reynolds. If that's the case, why did Hamilton do it himself, voluntarily destroying his career, personal life and public persona? It doesn't make sense to publicly publish something that could destroy your whole life. So why'd he do it?
Hamilton only wrote the pamphlet after the scandal had already come to light. It was published by Callender in a series of pamphlets purportedly about the early history of the US, and Hamilton suspected - probably with good reason - that the publication was encouraged by Madison and/or Jefferson. He then wrote the pamphlet to demonstrate that, while he had been unfaithful he had not been corrupt.
Source - Hamilton by R. Chernow