In the case of New York Times v. Sullivan, the Supreme Court was saying that there is a difference in what constitutes libel if you are talking about a public official as opposed to if you are talking about a private citizen. It is much easier to be found guilty of libelling a private citizen and much harder to be found guilty of libelling an official.
In order to be found guilty of libelling an official, a newspaper has to be guilty of "actual malice." This means that they have to know what they are printing is false or they have to have "reckless disregard" for whether it is true or not. It is very difficult to prove actual malice because it is very difficult to know what the newspaper publishers actually knew or intended when they published a piece.
In contrast, there is a much lower standard of proof if you are a private citizen and you claim you've been libelled. All you have to prove is that your reputation has been harmed -- nothing about what the person who published the libel knew when they published.