A standard example of two CW complexes which have isomorphic homotopy groups but are not homotopy equivalent is $ RP^2 \times S^3$ and $RP^3 \times S^2$.
The easiest way to see that they are not homotopy equivalent is by looking at their homology (i.e. ala the Kunneth formula).
What is an example of two CW complexes with isomorphic homotopy groups and homology, yet not homotopy equivalent?
It seems like an obvious place to look would be to have non-isomorphic cohomology (as rings). If there is such an example, then the next question would be for an example of two non-homotopy equivalent CW complexes with isomorphic homotopy groups, and isomorphic homology (as graded groups) and cohomology (as rings). Do such examples exist? I suspect they do, but I don't know any.
Update: there are good answers below to the first question, with links. But it seems to still beg the question about cohomology: does anyone know an example of two non-homotopy equivalent CW complexes with isomorphic homotopy groups, homology groups, and cohomology rings?