In a discounted payoff the borrower keeps the house. They come up with a lump sum, less than the full balance, from a refinance, a family loan, or savings, and you release the lien in exchange. The property doesn't change hands. You get cash, the borrower gets a clean title and stays put.
In a short sale the house is sold to a third party for less than the loan balance, and you accept the net sale proceeds as payoff. The borrower moves out. You're now dependent on a real estate transaction closing, so you're waiting on a buyer, an appraisal, an inspection, and a listing period.
Which one you aim for depends on what the borrower can access. A borrower with income and a relative willing to help is a DPO candidate. A borrower who has decided to leave and has equity above your discounted basis is a short sale candidate. You often find out which by making contact and asking, which is why borrower communication drives your outcome as much as the numbers do.
Two practical points. A DPO usually closes faster and cheaper because there are no agent commissions and no buyer financing to fall through, so the same dollar amount is worth more to you. And forgiven mortgage debt can have tax consequences for the borrower, and the reporting requirements on the holder of the debt are specific, so run any settlement past a tax professional before you sign a release. That's true of both paths and it's the part people discover afterward.