Applications of cohomology rings

It was recently asked on math.stackexchange ‘what are cohomology rings good for?’. As the answers there show, the ring structure is, in some sense, a ‘finer’ invariant then the group structure and can be used to prove that spaces which have the same cohomology groups, do not have the same homotopy groups (without resorting to higher homotopy, which is hard!)

Question: Show that $S^1 \vee S^2 \vee S^3$ and $S^1 \times S^2$ do not have the same homotopy type.