Hmm... tricky question. Formally it must use a preposition, but in everyday use we omit it for this verb. So, if this duo's question is a formal question, we will read the a as a preposition and there will be no article (it's facultative, and formal register tends to omit it more often). But it can be viewed as an informal question also, then the a will be an article. Il try to summarize it:
Ele responde (a) minha carta: informal. It has no preposition and the article is facultative
Ele responde a/à minha carta: formal. It has preposition and the article is facultative
It is preposition. See here - http://dictionary.reverso.net/portuguese-english/responder
If we use the English sister word, respond then it becomes in the translation, He responds to my letter so I would peg it ("a") as a preposition in Brazilian forms, and that in Euro PT it should be, Ele responde à minha carta (with the contraction of the preposition + article which is less optional outside of Brazil).
Great question (made me think, and I like that)! :)
The preposition is necessary here in the English (as with the Portuguese) when using "respond" but not with "answer".
- He responds to my letter
- He answers my letter
As to the differences between the two in regards to meaning, there is little here but, in other circumstances, responding does not always mean answer. For instance: "I have a question" "Okay, what is your question?" For this the response was a question and not an answer.