Should one provide a reference list and a bibliography in an academic paper? I was wondering whether it is good practice to present a reference list (works one actually cites) as well as a bibliography (additional literature that has been used but that is not referred to). Is there any difference between MLA and APA in the matter?

Both MLA and APA go quite deeply into how to construct a reference list, but a bibliography is seldom mentioned.
Typically, academic papers have traditionally provided the lists of works referenced, rather than bibliographies. Primarily, this is a matter of space, as well as recognizing work that has actually been used. Otherwise, it's too easy to include a bunch of random entries that have nothing to do with the paper in question. (It's still possible to do so in a works cited list, but you at least have to make the effort to work it into the paper somehow.)

In general, you should only include both lists if the journal in question allows such behavior. If you are publishing the paper in a venue that does not have such requirements, then you are of course free to include both lists if you choose. (Although I would not repeat the references list in the bibliography list, again to save space.)