Usual homotopy-type invariants of the configuration spaces may be able to distinguish between spaces which are homotopy equivalent but not homeomorphic. For instance, Riccardo Longoni and Paolo Salvatore [Configuration spaces are not homotopy invariant, Topology 44 (2005), no. 2, 375–380; MR2114713] distinguish between the lens spaces $L(7,1)$ and $L(7,2)$ by considering the universal covers of the (ordered) configuration spaces of $2$ points on these lens spaces, and showing that a certain Massey product vanishes for $\widetilde{F_2(L(7,1))}$ but not for $\widetilde{F_2(L(7,2))}$.

Usual homotopy-type invariants of the configuration spaces associated to a given space may be able to distinguish between spaces which are homotopy equivalent but not homeomorphic. For instance, Riccardo Longoni and Paolo Salvatore [Configuration spaces are not homotopy invariant, Topology 44 (2005), no. 2, 375–380; MR2114713] distinguish between the lens spaces $L(7,1)$ and $L(7,2)$ by considering the universal covers of the (ordered) configuration spaces of $2$ points on these lens spaces, and showing that a certain Massey product vanishes for $\widetilde{F_2(L(7,1))}$ but not for $\widetilde{F_2(L(7,2))}$.