befugt
authorized
Having official permission or the right to do something.
Nur der Geschäftsführer ist befugt, solche Verträge zu unterschreiben.
Only the managing director is authorized to sign such contracts.

authorized
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authorized
Having official permission or the right to do something.
Nur der Geschäftsführer ist befugt, solche Verträge zu unterschreiben.
Only the managing director is authorized to sign such contracts.

'Befugt' (authorized) is very often used in the construction 'befugt sein, etwas zu tun' (to be authorized to do something). It is therefore followed by an infinitive with 'zu' that describes the specific action for which the permission applies.
Der Sicherheitsdienst ist befugt, die Taschen zu kontrollieren.
The security staff are authorized to check the bags.
This word belongs to formal language and is used mainly in professional, legal, or administrative contexts. In everyday language, people would more likely use 'dürfen' (to be allowed to) or 'die Erlaubnis haben' (to have permission).
Laut Vertrag ist er nicht befugt, Untermieter aufzunehmen.
According to the contract, he is not authorized to take in subtenants.
'Befugt' (authorized) belongs to the word family of 'die Befugnis' (authority, authorization, competence). The opposite is 'unbefugt' (unauthorized), which is often found on warning signs (e.g., 'Unbefugten ist der Zutritt verboten' — access is prohibited to unauthorized persons).
Das Betreten der Baustelle ist für Unbefugte strengstens verboten.
Entering the construction site is strictly prohibited for unauthorized persons.
responsible
'zuständig' and 'befugt' both have to do with official roles. 'zuständig' means that someone is responsible for a task or area. 'befugt' means that someone officially has the permission or right to do something. The words can be confused because a person who is responsible is often also authorized. But they are not always the same: you can be responsible for a task but not be allowed to make certain decisions. 'zuständig' fits when talking about responsibility, while 'befugt' fits when talking about permission or decision-making authority.

Ihr seid nicht befugt, nach der Arbeit die gesperrten Dateien zu öffnen.
You are not authorized to open the restricted files after work.
Im Gespräch erklärt sie, wer für die Vorbereitung des Festes zuständig ist.
In the conversation, she explains who is responsible for preparing the festival.