The person in the old testament would have sins remitted but not redeemed. Jesus redeemed all sins that were only remitted that needed redeeming at the cross.
hebrews 9.15
15And for this cause he is the mediator of the new testament, that by means of death, for the REDEMPTION of the transgressions that were under the first testament, they which are called might receive the promise of eternal inheritance.
Rom 3 24Being justified freely by his grace through the REDEMPTION that is in Christ Jesus:
25Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God;