$stmt->free_result() frees up memory related to a result set, whereas $stmt->close() frees up memory related to a prepared statement. Subsequently calling $stmt->close() will cancel any result still remaining.
In essence, calling $stmt->close() will provide the same effect as calling $stmt->free_result() since it cancels the result set as well. But calling $stmt->free_result() will not clear out the memory used by the prepared statement in which case you must use $stmt->close().
As far as which one to use goes - there may be situations where you
intend on using the prepared statement you have initialized, but are no
longer in need of the result set you currently have. In such a case you
would wait on calling $stmt->close() until you are done with the prepared statement and instead call $stmt->free_result() before executing another statement.