The number one reason that we advocate pursuing a short sale vs. a foreclosure, is that a foreclosure will prevent you from obtaining a mortgage for a minimum of five to seven years, in addition to extensive damage to your credit, whereas a short sale will have far less impact on your credit in that most borrowers will be able to obtain a mortgage after two years of conducting a short sale.
Also, the deficiency (or tax consequences) in the event of foreclosure, if is collectable, will be significantly higher than in a short sale (since properties sell at extremely discounted prices at foreclosure auctions).
The second reason that we advocate short sales is that promissory notes and deficiency judgments before and after the sale are, for the most part, negotiable. In many cases, the deficiency owed can be negotiated to a percentage (ie. 10% of a Bank of America HELOC loan) or sometimes completely waived.
The lender may forgive the balance in exchange for a small pay-off or an affordable payment arrangement with the borrower. This largely depends on lender policy and type of loan. Sometimes, it is even possible to have the buyer pay the difference!