What happens to money paid pursuant to a conviction that was vacated on appeal?

The State of Ohio contends that the answer depends on precisely what the purpose of the payment was.

Victor Brown was convicted in 2010 of failing to register his address in connection with a gross sexual imposition offense that had been committed in 1993. He was sentenced to six months imprisonment and ordered to pay about $250 in court costs. His conviction was later vacated, but money to cover the costs was periodically deducted from his prison account anyway. He moved for an order ceasing the garnishments and refunding the costs already deducted, which was denied.

On appeal, the State conceded that Brown was entitled to a refund of the court costs paid, and so the court of appeals agreed.  But interestingly, the State's brief asserts that if the money paid had been part of a restitution award, then the defendant would not have been entitled to a refund, even if the judgment ordering restitution had been vacated.

Perhaps surprisingly, a very similar issue was before the US Supreme Court just this week. In Nelson v. Colorado, the question presented asks whether the State of Colorado can require defendants whose convictions have been overturned to prove their actual innocence, in a separate proceeding against the state and by clear and convincing evidence, before being entitled to a refund. A decision in Nelson is expected this spring, but put me down in the camp that thinks that the petitioners will prevail.

(Update: The petitioner in Nelson did indeed prevail, with the Supreme Court ruling 7-1 that the Colorado statute at issue was unconstitutional.)