It all gets pretty defintional. In other words only have whatever meaning we give them.
But if you're comparing "Jewish delicatessen" corned beef and pastrami, pastrami is NOT smoked corned beef.
Corned beef takes on a lot more salt and "cure" during the brining process. It is cooked by steaming. Traditionally it is made with the flat only, and trimmed fairly lean. Or, as the saying goes, "not too fat."
Pastrami is also cured and brined, but not made nearly as salty. After the brning it is dry rubbed with a sugar and spice (lots of pepper) mixture before smoking. Finally it is smoked. Pastrami can be made with the point or the flat. In the last forty years, or so, fat has gone out of fashion, so "high end" pastrami is usually made with the flat.
Because the brisket fat cap doesn't render and isn't particularly palatable, meat must be fairly closely trimmed before being pastramied (yes, it's a word), if you want to get flavor on both sides. I go for an 1/8" trim, taking it down almost to the meat, leaving plenty of streaks of red. That's okay though. Because it's been brined it will stay fairly moist. And because the bark isn't important, there's no problem with wrapping after you've got enough smoke in it. The wood of choice is cherry.
The traditional method is to smoke, then finish in the oven. A lot of barbecuers consider this heterodox to the point of anathema. Personally, I subscribe to the "whatever works for you" school.
One thing that's for sure, when you carve, cut slices as thin as will still hold together. Of course, carve against the grain. This brings out the play between the rub and the meat, as well as making for a more tender bite.
SoEzzy's suggestion about soaking the salt out of a corned beef is valid. However, be aware that the long brine and cure has altered the brisket's cell structure and that in order to get the salt out, you are further diluting the meat's natural juices -- already diluted in the brine. In other words, you can do it and even do a pretty good job of it if you're REALLY jonesing for a brined brisket to make pastrami. But, you can do a much better job from scratch if you've got access to decent brisket and the time to cure.
The one time it really makes sense if you've picked up a ton of high quality buck a pound brisket during the St. Paddy's Day sales. At some point you've got to think seriously about moving it out of your refrigerator and freezer. Yes, even the ones in the garage and on the patio. Unfortunately that point comes at the same date you've eaten enough corned beef to last a life time.
JB has my pastrami recipe and is threatening to make some long before I'm going to get around to it again. So we should hear something reasonably soon.
Not too lean,
Rich