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I believe the dry chorizo would look more like pepperoni where as the regular chorizo would be more like sausage. So yep they are different. I would imagine cooked regular chorizo would be quite tasty in the sandwich as well.
The chorizo used in this recipe is the cured, Spanish chorizo. The other style of chorizo is the Mexican, which is fresh, uncooked, and in casings. (links) The flavors are quite different, but both can be quite wonderful.