In a short sale the seller is disclosing that the proceeds from the sale will fall short of the required closing costs. Most lenders will not start the short sale approval process until they have an offer. Many times that means that if you are the first accepted offer, THEN they start the approval process., it could take a long time for them (six months) to approve the sale.
So when you say "approved short sale", I would assume that the process will proceed at a more normal pace (a month or two).
Looks like TJ and Beth have both provided good answers. My understanding of 'bank approved short sale' is that the lender has told the owners a number that they would be willing to accept on the sale of the property. Whereas a 'short sale' inplies that the seller is willing to accept less that they owe on the property, but they do not have from the lender, the dollar amount that the lender would be willing to accept on the sale.