But what exactly is the difference between POST and SEARCH? Both include the request parameters in the body, so they would be obscured from the user. Unless they aren't, in which case it is a matter of the choices made in the implementation.

Is it implied idempotency and the lack of a confirmation dialog when the user reloads the page?

Cachability, perhaps. You cannot cache a post, but may be able to cache a search or query verb.