In New England the word is grinder, and the word carries a build, not only a regional accent. Elsewhere on the long-roll map the local name is mostly nominal: a hoagie and a hero are the same cold Italian idea wearing different town labels. Across much of New England the grinder defaults hot. Order one and the expectation is a roll that has been to the oven: meats, cheese, and sauce run under heat until the cheese melts and the crust toasts. The name is the most local thing about it, and what it specifies is heat. That is the distinction worth drawing, not the generic anatomy of a sub, which the long-roll family covers elsewhere.
The craft is in a roll chosen to be heated rather than just filled. A New England grinder roll is a touch sturdier and crustier than a soft sub roll because it has to spend time under a broiler or in a pizza oven without going brittle, and it has to absorb a hot, often saucy filling and still lift in one piece. The cold Italian grinder exists and follows the shingled-and-dressed logic of any long roll, but the builds the word is most associated with are the hot ones: a steak and cheese, a meatball, a chicken or eggplant parm, where the roll is a vessel for a wet filling and the quick toast is what keeps the crumb from surrendering to the sauce before the last bite. The cheese is added before the heat so it binds the filling to the bread as it melts rather than sitting on top. Many New England grinders come out of pizzerias, which is why the oven, not a flat-top, is the defining tool, and why the line between a grinder and a sub in this region is drawn at the oven door.
The variations stay close to that oven. A cold Italian grinder dressed with oil and shredded lettuce, a hot pastrami or steak-bomb build, a parm run on the same toasted roll. The wider long-roll shelf, the New York hero, the Philadelphia hoagie, the chain submarines, runs the same architecture under other names. Those deserve their own articles rather than being crowded in here.