It is a standard consequence of Hurewicz's theorem that a homology eqivalence between simply connected spaces is a weak equivalence (and hence a homotopy equivalence, if the spaces are CW-complexes).
What is more, it is even enough to assume that the map is a homology equivalence with local coefficients and an iso on $\pi_1$, see e.g. this question.
(On the other hand, a map which is only a homology equivalence does not need to be an isomorphism on $\pi_1$ and hence not a weak equivalence.)
What is a concrete example of a map that is an isomorphism on homology with $\mathbb Z$ coefficients, and also an isomorphism on $\pi_1$, but not a weak equivalence?