
antworten
to answer
To give an oral or written response to a question, a message, or a statement.
Bitte antworten Sie auf meine E-Mail bis spätestens morgen.
Please answer my email by tomorrow at the latest.
to answer / respond
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to answer
To give an oral or written response to a question, a message, or a statement.
Bitte antworten Sie auf meine E-Mail bis spätestens morgen.
Please answer my email by tomorrow at the latest.

to respond
To react to an action, event, or stimulus. This meaning is somewhat more figurative than a direct answer to a question.
Die Mannschaft antwortete auf den Gegentreffer sofort mit einem eigenen Tor.
The team immediately responded to the goal they conceded with a goal of their own.
The verb 'antworten' (to answer) belongs to the noun 'die Antwort' (the answer). The prefix 'ant-' originally means 'against' or 'in response to.' So an answer is literally a kind of 'counter-word' to a question.
Auf meine Frage bekam ich eine klare Antwort.
I got a clear answer to my question.
'Antworten' (to answer) always needs the preposition 'auf' with the accusative when you say what someone is answering. A direct accusative is wrong. Don't say: 'Ich antworte die E-Mail.'
Er hat immer noch nicht auf meine Nachricht geantwortet.
He still hasn't answered my message.
The person you answer is in the dative. You can combine both constructions: 'jemandem (Dativ) auf etwas (Akkusativ) antworten' (to answer someone about something).
Könntest du mir bitte auf meine Frage antworten?
Could you please answer my question?
In the meaning 'reagieren' (to react/respond), 'antworten' is also used with 'auf + Akkusativ.' The context makes it clear that this is not a verbal answer, but a reaction to an action.
Die Firma antwortete auf die Kritik mit einer öffentlichen Entschuldigung.
The company responded to the criticism with a public apology.
to answer
Both words refer to a spoken or written answer to a question or message. „Beantworten“ takes the question or message directly as an accusative object, while „antworten“ is generally used with „auf“ or with a person in the dative. The meaning is similar, but the sentence structure determines the phrasing.

Ich antworte dir auf deine Frage nach dem Termin heute noch.
I’ll respond to you about your question regarding the appointment later today.
Ich beantworte deine Frage nach dem Termin heute noch.
I’ll answer your question about the appointment later today.