What is the difference between Authentisierung and Authentifizierung in German security terminology?

Authentisierung = the user proves who they are (active, by the user). Authentifizierung = the system verifies the proof (active, by the system). Side-effect of being in German — both translate to "authentication" in English.

The relationship:

• Benutzer authentisiert sich am Server — the user provides credentials (presents themselves).
• Server authentifiziert den Benutzer — the server checks those credentials and decides yes/no.

Benutzer ──[authentisiert sich am]──→ Server
Benutzer ←──[authentifiziert den]──── Server

Why the distinction:

• Strictly in German, these are two sides of the same handshake — the act of proving and the act of verifying.
• In day-to-day English we say "the user authenticates" and "the server authenticates the user" using the same verb — the context disambiguates.
• German Wikipedia treats them as a matched pair: every authentisierung (the user's act of presenting proof) needs an authentifizierung (the verifier's act of checking it) on the other end.

Tip: when you read German security material, authentisierung is almost always what the user/client does, and authentifizierung is what the server does. They're not synonyms; they're the two directions of the same protocol step.