A fair use defense means that something happened that would otherwise create liability, but doesn't because of some (often policy) reason. Making something an affirmative defense has procedural implications. But "rights" can be implemented as affirmative defenses. For example, truth is an affirmative defense to defamation in the U.S., even though it's closely connected with free speech rights.

EDIT: But see dragonwriter's very pertinent comment about defamation cases involving matters of public concern, where falsity is an element of the claim rather than an affirmative defense.

The practical distinction is that it's easier to dispose of cases on a motion to dismiss when something is a required element of the claim versus when it is an affirmative defense.