They mean the same thing. "Don't respond" sounds more formal, but either could be used by native English speakers.

'Answer' is is similar to 'reply' - it's usually a yes or a no, or the information which the other person has requested. You give answers when someone has asked a question or requested information. 'Respond' is broader than 'answer'. It doesn't even have to be verbal. You could respond with an action, such as laughing, hitting someone, kissing them, or walking out of the room, for example.