Mortgage and a deed of trust are not the same even though people usually use the term "mortgage" interchangeably for both.
Mortgages involve two parties, borrowers and lenders; while deeds of trust have third parties, called trustees, who hold temporary title to the properties until borrowers pay off their loans.
That difference can be crucial when a borrower falls behind in payments. With deeds of trust, the trustees don't have to go to court to initiate a foreclosure; with a mortgage, the lender almost always has to go to court, which slows down the process. In states where deeds of trust are an option, lenders almost always choose them over mortgages, because they are "non-judicial" - and quick.
In contrast, judicial foreclosure, which is the usual procedure when a mortgage is involved, can be slow. First a lawsuit must be filed. Then, there is a period of discovery and a court date must be set. And courts can grant delays to prepare cases. All told, it can take many months.
edit Rights of Redemption
Another difference is that liberal "rights of redemption" are more common in "deed of trust" states than in mortgage-only states. That means even if your home is foreclosed on and auctioned off, there is a time period when you can pay the debt and get the home back. This window is usually very brief, but can last for as long as a year after the property is sold at auction.
The right of redemption can be tough on foreclosure buyers. The state gives the owner the right to buy back the property for the price the auction bidder paid for it. Any additional money that buyers paid, such as for repairs or maintenance, is lost.
edit Prevalence in U.S. states
Deeds of trust are the more common of the two, used in 34 states either mostly or exclusively. States such as California, Alabama, New Hampshire, Mississippi, Michigan allow use of deeds of trust.
States such as Vermont, Florida, Nebraska and New York are examples of mortgage-only states.
edit Video explaining the differences