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Corned beef is the "flat" portion of the brisket that has been cured in a brine of salts and spices to give it it's great taste. It is then traditionally boiled or braised in order to cook it.
Pastrami is a corned beef that has been taken one step further in that it has been smoked in order to cook it.
Both are excellent and a very tender result from a tough cut of meat.
Corned beef can be made from several cuts of beef. Many of the commercial ones are made from inside round. Pastrami is a corned beef that is spiced in pastrami fashion and smoked. The same applies to montreal style smoked meat. I make them all the time with bear roasts.
Corned beef is generally a cut of brisket that has been cured, corned refers to the fact that they used to rub large granules of salt into the meat that they called "corns".
When you take your meat and cure it it is corned beef, once you place it in the smoker and cook it it is pastrami, some will argue it must be seasoned as well, I tend to agree but if you don't cook with smoke then it technically wouldn't be pastrami.
Got a nice chipotle pastrami in the smoker right now as a matter of smoke