Is there any difference between "schrijfster" and "schrijver"?
Yep. A 'schrijver' is a male writer and a 'schrijfster' is a female writer.
So why isn't "de schrijster verkoopt haar boeken" accepted?
Maybe Duo expects the male form. Or it's the typo (it's schrijfster).
The typo I made only in the comment, and Duo marked as the only mistake the word "haar".
It clearly says his books.
Right, but then it should mark as a mistake also the word "schrijfster", because, even though "writer" in English is not gender-marked, the sentence is referring to a male writer, thus, the word "schrijfster" should me marked as wrong, too.
No, let's assume it's a female writer. She can sell someone else's books, in this case his books. There's nothing that says that the books were written by the writer.