English often takes words with the same meaning from different places, even though the basic meaning is the same. The similar words to 'response' from other European languages are from the Latin root, "responsum" (to answer). For 'answer', you'll find parallels in the Germanic languages ('answer' meaning 'to swear against [something]'). As Jura points out, there's no real difference these days. However, for more formal speech the Latin-based "response" is preferred.

The meaning is the same.

repons refers to: 1 A result "this situation developed in response to events in Africa" 2. A bodily process occurring due to the effect of some antecedent stimulus or agent. "his responses have slowed with age" .3. A statement (either spoken or written) that is made to reply to a question or request or criticism or accusation 4. The manner in which something is greeted 5.A phrase recited or sung by the congregation following a versicle by the priest or minister 6.The speech act of continuing a conversational exchange 7. The manner in which an electrical or mechanical device responds to an input signal or a range of input signals While answer refers to 1.React verbally She didn't want to answer"; "answer the question"; "We answered that we would accept the invitation" 2.Respond to a signal "answer the door"; "answer the telephone" 3.Give the correct answer or solution to "answer a question"; "answer the riddle" 4.Understand the meaning of The question concerning the meaning of life cannot be answered" 5. Give a defence or refutation of (a charge) or in (an argument) "The defendant answered to all the charges of the prosecution" 6. Be liable or accountable "She must answer for her actions" 7. Be sufficient; be adequate, either in quality or quantity "A few words would answer" 8.Match or correspond The drawing of the suspect answers to the description the victim gave" 9.Be satisfactory for; meet the requirements of or serve the purpose of "This may answer her needs" 10React to a stimulus or command "The steering of my new car answers to the slightest touch" cowboy...