The distinction is that the question of whether the publisher harbors ill will is never directly addressed. A publisher that likes the defamed person doesn't get off because they like them.
-- The very definition of malice, in defamation law as well as elsewhere, is ill will, including reckless disregard. --
The question of ill will does not appear in NYT v. Sullivan. The definition of "actual malice" in that case is knowing the statement is false, or having reckless disregard for the truthfulness of the statement. Now, one might say that reckless disregard can ONLY occur in combination with or motivated by ill will.
Interestingly, NYT's preliminary defense, in its answer to Trump's lawyer's demand letter, was that the accusations do not harm Trump's reputation.
Fine points about the intent and recklessness of malice. Suffice to say, in a defamation suit, the malice of recklessness or intent is enough. The acts is one of ill will which is sufficient, even if the thought isn’t.