Schnaps (European version) is typically presented as an unaged fruit brandy that is strong and dry. In Germany, Schanps are often consumed before, during, and after meals to aid with digestion. They are enjoyed at room temperature in small glasses with the serving drank quickly and all at once.
Schnapps (American version) are fruity, herbal, and infused with sweeten grain alcohol. They are often used in cocktails or served as mixers. They are lower in ABV and are typically sweet.
Bierschnaps as the name might suggest, is more of a European thing. This is an unaged spirit made from distilled beer. You might see it as hop schnaps. Instead of originating from fruit, this comes from beer, which is at least 51-100% malt based.