OK, maybe this is right and may help things along (I did say maybe)
"Brisket" is the name of the cut of beef from like the front breast area of a bovine
"Corned" is some bastardization of "brined", soaked in salt to preserve and flavor
The typical or traditional use of the term "corned beef" refers to a peice of meat, usually brisket or a part of the brisket, that has been brined for a period of time and then boiled or steamed with or without other veggies such as cabbage or potatos.
The typical use of the term "Pastrami" refers to a simular cut of meat that has been brined in a simular fashion as the corned beef but usually with more pepper or slightly different spices or an added surface rub of spices after the brining process and then smoke cooked.
Sometimes Pastrami, and maybe even corned beef, are not brined in a solution but rather "Dry cured" by applying the salt and spices to the out side of the meat and left to cure. ( I am not so sure about this, it may have been a traditional method, but I am thinking that a solution brine is the modern way)
Typically, the corned beef would be cooked for a special occasion (St.Pattys or such) and served hot out of the pot/oven and consumed the same day that it was cooked.
Pastrami on the other hand tends to be brined and smoked and then put in reserve and eaten in a more drawn out process as a typical lunch meat sliced thin while stiff from cold or dry storage. A treat is a "Hot Pastrami" sandwich where some of the lunch meat is sliced and heated for the sandwich. There are howevr occasions and those that serve pastrami by heating the entire pastrami by steam for a couple of hours and then slicing it thicker and serving it on a plate simular to corned beef.
You can smoke a store bought corned beef and get a version of pastrami that is quite tasty, but I would reccomend starting out raw and brining your own and smoking it yourself. There are some subtle differences in the spices and you may have better control or selection of the spices and the choice of meat.
Yummmmmm, puhhhsrtammmmmeeee, I think I know what my next project will be
There are many post around here about the patrami side of the corned beef adventure, hunting them and reading them is an adventure all its own!