Yes, it's the present perfect continuous. The use of the present perfect continuous focuses on a more extended period of time than the present perfect simple, and on the repetition of actions during that period. If you say 'Why haven't you answered my letters?', you are concentrating on one action - on the one thing which the other person has so far failed to do. The implication is that you only expected one reply, but you haven't received this. 'Why haven't you been answering my letters?' focuses on a whole period of time, and the events of this period. During this period you have been sending letters repeatedly and the other person has repeatedly been ignoring them. The implication is that you expected the person to reply to each of your letters.

"Why haven’t you been answering my letters?" sounds as if many letters were sent.

Maybe using a different verb for explaining the differences of the two tenses/aspects is easier: Please allow me to use "to read". I have read the book. (present perfect) --> This means: I read the book sometime before I said this sentence. I will have been reading the book. (present perfect continuous) --> This sentence is structured with: will (future) + have been (perfect) + be_Ving (progressive, or continuous). It means: Sometime in the future, say, 2:00 tomorrow afternoon, I will have read some of the book and I will continue reading the book at that time.

I'm not a professional, but I would say the difference is in the active and not active voice. One example uses the be verb and there are changes to the ending for answering and answered. "You not been answering." (not that you would actually say it like that), and "not answered" The first is recent past and the second is anytime in the past.