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A fairly easy obstruction is the action of the fundamental group on the homotopy groups. Here is an example where the homotopy groups are isomorphic as abstract groups, but not preserving the action of the fundamental group.

Form a fibration over a circle with fiber a wedge of spheres $S^2\vee S^2$. That is, take the wedge, cross with an interval, and identify the ends by a homotopy equivalence of the wedge. Homotopy equivalences are parameterized by $GL_2(\mathbb Z)$, the action on homology. The homotopy type of the space remembers the monodromy as the action of the fundamental group on the homotopy groups. The homotopy groups are that of the universal cover, which does not depend on the choice of monodromy. Compute the homology by the Serre spectral sequence. This involves the homology of $\mathbb Z$ acting on $\mathbb Z^2$ by the monodromy. If the monodromy is hyperbolic, the homology vanishes and the space has the homology of the circle. Thus two different hyperbolic matrices give spaces with isomorphic homomotopy homotopy and homology groups.

In a completely different direction, there are examples of pairs of simply connected spaces such that the Postnikov truncations are equivalent, but the inverse limits of their Postnikov towers are not, but I think such examples have to be pretty large.

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A fairly easy obstruction is the action of the fundamental group on the homotopy groups. Here is an example where the homotopy groups are isomorphic as abstract groups, but not preserving the action of the fundamental group.

Form a fibration over a circle with fiber a wedge of spheres $S^2\vee S^2$. That is, take the wedge, cross with an interval, and identify the ends by a homotopy equivalence of the wedge. Homotopy equivalences are parameterized by $GL_2(\mathbb Z)$, the action on homology. The homotopy type of the space remembers the monodromy as the action of the fundamental group on the homotopy groups. The homotopy groups are that of the universal cover, which does not depend on the choice of monodromy. Compute the homology by the Serre spectral sequence. This involves the homology of $\mathbb Z$ acting on $\mathbb Z^2$ by the monodromy. If the monodromy is hyperbolic, the homology vanishes and the space has the homology of the circle. Thus two different hyperbolic matrices give spaces with isomorphic homomotopy and homology groups.

In a completely different direction, there are examples of pairs of simply connected spaces such that the Postnikov truncations are equivalent, but the inverse limits of their Postnikov towers are not, but I think such examples have to be pretty large.