One way to think about how to distinguish, if not classify, such spaces is by <a href="http://ncatlab.org/nlab/show/primary+homotopy+operation">homotopy operations</a>. In the same way that cohomology operations are natural transformations between cohomology functors, homotopy operations are natural transformations between homotopy functors. By the Yoneda lemma, natural transformations $\pi_n \to \pi_m$ are in natural bijection with the homotopy group $\pi_m(S^n)$, so elements of the homotopy groups of spheres give unary operations on homotopy.

More generally, the $k$-ary operations on homotopy groups are natural transformations $\pi_{n_1} \times \dots \times \pi_{n_k} \to \pi_m$, and by the Yoneda lemma these are in natural bijection with the homotopy group $\pi_m(S^{n_1} \vee \dots \vee S^{n_k})$. For example, the <a href="http://en.wikipedia.org/wiki/Whitehead_product">Whitehead bracket</a> $\pi_n \times \pi_m \to \pi_{n+m-1}$ is a well-known family of binary operations coming from some distinguished homotopy classes of maps $S^{n+m-1} \to S^n \vee S^m$. They make the homotopy groups of a space into a graded Lie algebra (up to a degree shift).

A product of Eilenberg-MacLane spaces always has trivial homotopy operations, so you can distinguish a space from a product of Eilenberg-MacLane spaces by checking to see whether any of its homotopy operations are trivial. For example, the homotopy operation $\pi_2(S^2) \to \pi_3(S^2)$ given by a generator of $\pi_3(S^2) \cong \mathbb{Z}$ must be nontrivial since it is the universal example; this shows that the $3$-truncation of $S^2$ is a homotopy type with $\pi_2 \cong \pi_3 \cong \mathbb{Z}$ and all other homotopy groups trivial but which cannot be homotopy equivalent to $B^2 \mathbb{Z} \times B^3 \mathbb{Z}$. 

Moreover, it's a classical result that $H^4(B^2 A, B)$, the cohomology group that classifies the $k$-invariant of a space with $\pi_2 \cong A, \pi_3 \cong B$, can be identified with the group of quadratic functions $A \to B$. Given such a $k$-invariant, the corresponding quadratic function turns out to be precisely the homotopy operation $\pi_2 \to \pi_3$. So in this case the classification by $k$-invariants and by homotopy operations agree.