For any $n > 2$, $\mathbb{CP}^n/\mathbb{CP}^{n-2}$ and $S^{2n}\vee S^{2n-2}$ have the same cohomology groups: for any ring $R$, we have
$$H^k(\mathbb{CP}^n/\mathbb{CP}^{n-2}; R) \cong H^k(S^{2n}\vee S^{2n-2}; R) \cong \begin{cases} R & k = 0, 2n-2, 2n\\ 0 & \text{otherwise}. \end{cases}$$
Moreover, the two spaces have the same cohomology ring structure (the product of any two elements of positive degree is necessarily zero).
However, for $n$ even, the two spaces are not homotopy equivalent. As is discussed here and here, this can be shown by demonstrating that $\operatorname{Sq}^2 : H^{2n-2}(\mathbb{CP}^n/\mathbb{CP}^{n-2}; \mathbb{Z}_2) \to H^{2n}(\mathbb{CP}^n/\mathbb{CP}^{n-2}; \mathbb{Z}_2)$ is an isomorphism, while $\operatorname{Sq}^2 : H^{2n-2}(S^{2n}\vee S^{2n-2}; \mathbb{Z}_2) \to H^{2n}(S^{2n}\vee S^{2n-2}; \mathbb{Z}_2)$ is the zero map. In particular, we see that the cohomology of the two spaces are not isomorphic as modules over the Steenrod algbera.
I am looking for a similar example where the spaces are closed manifolds.
Are there examples of closed manifolds with isomorphic cohomology rings, but different cohomology modules over the Steenrod algebra?