There are such spaces, for example X = S^2 \times RP^3, Y = S^3 \times RP^2.
(These are both smooth and CW-complexes.)

Whitehead's Theorem says that for CW-complexes, if a map f : X \to Y induces an isomorphism on all homotopy groups then it is a homotopy equivalence.  But, as the example above shows
you need the map.

(Whitehead's Theorem is not true for spaces wilder than CW-complexes.  The Warsaw circle
has all of its homotopy groups trivial but the unique map to a point is not a homotopy equivalence.)