Hopefully this question is not too vague to be closed. I am looking for examples of when a construction/theorem that involves $E$-(co)homology or even simply the ring $E_*$ requires an understanding of an some sort of $E_n$-structure on $E$ or its module category. For example:
When $n=\infty$, $E_\infty$-rings possess the power operations $P_n:E^*(X)\to E^*_{\Sigma_n}(X)$.
When $n=3$, and $X$ is a quasi-projective scheme, $Br(X)\to H^2(X; \mathbb{G}_m)$ is an isomoprhism, but not in general. However, if $X$ is quasi-compact and quasi-seperated (or more generally, a derived scheme based on non-connected ring spectra with the same conditions, then $Br_{der}(X)=\pi_0(\mathfrak{br}(X))\to H^2(X; \mathbb{G}_m)$ is an isomorphism.
The construction of $Tmf$ naively is only an inverse limit over the category of the Landweber spectrum coming from elliptic curves. This limit though cannot be computed until coherence conditions are fixed, and this is only possible by noting that the subset of $E_\infty$-maps between these lifts is far more manageable.
Similar statements for Picard groups and other invariants exist. I am looking for more examples of situations where the normal algebraic category of $E_*$ ($E^*$)-modules doesn't cut it, but interesting results can be obtained only by working in a structured category.
