S<sup>3</sup> x RP<sup>2</sup> and RP<sup>3</sup> x S<sup>2</sup> are both smooth 5-manifolds with fundamental group Z/2 and universal cover S<sup>3</sup> x S<sup>2</sup>, so their homotopy groups are all the same.  On the other hand, only the latter is orientable since RP<sup>3</sup> is orientable but RP<sup>2</sup> isn't, so they have different values on H<sub>5</sub> and therefore can't be homotopy equivalent.  (I think this example is in Hatcher somewhere.)