There has always been alot of bad blood between Snape and Black. After all Black sent Snape into the passage after Lupin, knowing what was at the other end. If it hadn't been for James Snape would have been killed. Although I think that if Snape had known that Black was innocent of the crimes he would have taken him to Dumbledor instead of handing him over to the Dementors. Of coarse he didn't know this. If it hadn't been for Harry Snape would have handed him over to suffer the death eaters kiss. Probably out of pure bitterness and hatred.