Anarchy, meaning no rulers, can be—and, as we’ve all seen, typically is—conceived as total lawlessness, chaos, riots in the streets, etc. And this is understandable, I think, because “no rulers” can easily include oneself, i.e., not ruling even oneself.
Autarchy, on the other hand, is defined as self-rule, while at the same time specifying “and ruling nobody else,” i.e., no (external) rulers. So this more aptly addresses both sides of the liberty coin—liberty, in the anarchistic sense, as well as the requisite personal responsibility. Self-rule, in my opinion, implies self-mastery, self-discipline, self-reliance, and even developing a healthy self-esteem—defined as the dual facets of self-effacy (competence, trust in one’s mind) and self-worth (belief that one deserves love and happiness). source
It's interesting that more people are starting to make this semantic, yet highly meaningful distinction.
I remember for a minute, back when people could still be famous for being atheists, that many of them said something to the affect that they preferred some other term than atheism altogether, as they didn't want to be defined by what they didn't believe.
On a crude level, opposing 'autarchy' to 'anarchy' is rather like that. It speaks more to what one practices rather than what one doesn't practice.
It's also free from the implication that if there are in fact rulers something must necessarily be done about it. The true autarch, on the other hand, will not wait for rulers to vanish before he rules himself and makes the most of his life. Nor does he nitpick about who owns what or who should own what. He is, inevitably, the natural leader who inspires others to lead themselves.