Braunschweiger and liverwurst are types of German liver sausage. They are both made with pork liver and sometimes contain beef liver as well. Braunschweiger and liverwurst, however, are not exactly the same thing. Traditionally Braunschweiger is smoked, whereas liverwurst typically is not.
Liverwurst is a general term referring to any liver sausage. Braunschweiger is a smoked liver sausage that is named after the town Braunschweig in Germany. Therefore, a Braunschweiger is a liverwurst, but not all liverwurst are Braunschweiger.
There is no specific recipe for either sausage, as they are traditional foods. Braunschweiger is often used as a sandwich spread. Liver sausage also can be smooth and spreadable or used as a paté.
Germans have been making sausage since at least the ninth century, although the Roman cookbook "Apicus Cookery," dating back to 228 A.D., indicates the food has made as long ago as the third century. The process of combining the leftover bits of meat into a single food was practical because no part of the animal went to waste, leaving more food for hungry people. Therefore, sausage is one of the many foods born out of necessity that has remained popular even during times in which food is abundant.