What goes wrong in the axiomatic definition of a generalized (co)homology theory if one drops the axiom of homotopy invariance i.e. that homotopic maps should induce the same map in (co)homology?

Or do we have examples? Are there "interesting" or "useful" functors $\mathfrak{h}^{\cdot}:\mathrm{Spaces}\to \mathrm{Ab}$ that respect all Eilenberg-Steenrod axioms except homotopy invariance? Could such an $\mathfrak{h}$ be used to distinguish between two homotopy equivalent non-homeomorphic spaces?

Or do we have examples? Are there "interesting" or "useful" functors $\mathfrak{h}^{\cdot}:\mathrm{Spaces}\to \mathrm{Ab}$ that respect all Eilenberg-Steenrod axioms except homotopy invariance?