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 https://mathoverflow.net/questions/3540/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.)