I never understood the distinction until it was explained to me in the most powerful way possible by an ex-Catholic Brother who, unbeknownst to me, had long ago lost his faith. We had never met and I was calling to tell him of the imminent death of a mutual friend. Thinking he was still involved with the Catholic Church, I explained that I was an atheist in an attempt to pre-empt any discussion of god he might initiate. I was stunned when he responded that he had become an anti-theist. I asked him to explain the difference, and his explanation is forever burned into my memory. He said “An atheist doesn’t believe in the existence of god; an anti-theist believes that the very belief in god and religion is a destructive force on society.”
So there’s that.