Is the distinction between quasi-isomorphism and `weak homotopy equivalence' ONLY that the first means inducing an isomorphism in homology and the second to an isomorphism of homotopy groups?
This is a terminology question.
I think that "weak homotopy equivalence" is mostly used for maps between topological spaces (as opposed to "weak equivalence", that is used in the much broader context of model categories).
The term "quasi-isomorphism", on the other hand, is typically used for (co)chain complexes, or (co)chain complexes equipped with extra structure.
So I would say that the main difference between those two terms is that they are used in different contexts.