A common caution about Whitehead's theorem is that you need the map between the spaces; it's easy to give examples of spaces with isomorphic homotopy groups that are not homotopy equivalent. (See Are there two non-homotopy equivalent spaces with equal homotopy groups?). It's surely also true that the pair (homotopy groups, homology groups) is not a complete invariant, but can anyone give examples? That is, I'm looking for spaces $X$ and $Y$ so that $\pi_n(X) \simeq \pi_n(Y)$ and $H_n(X;\mathbb{Z}) \simeq H_n(Y; \mathbb{Z})$ but $X$ and $Y$ are still not (weakly) homotopy equivalent.

(Easier examples are preferred, of course.)

A common caution about Whitehead's theorem is that you need the map between the spaces; it's easy to give examples of spaces with isomorphic homotopy groups that are not homotopy equivalent. (See Are there two non-homotopy equivalent spaces with equal homotopy groups?). It's surely also true that the pair (homotopy groups, homology groups) is not a complete invariant, but can anyone give examples? That is, I'm looking for spaces $X$ and $Y$ so that $\pi_n(X) \simeq \pi_n(Y)$ and $H_n(X;\mathbb{Z}) \simeq H_n(Y; \mathbb{Z})$ but $X$ and $Y$ are still not (weakly) homotopy equivalent.

(Easier examples are preferred, of course.)