Let $T$ be a real torus, and let $X$ and $Y$ be $T$-spaces. Under what conditions (if any) will the existence of graded $H^*_T$-algebra isomorphism between the $T$-equivariant cohomologies of $X$ and $Y$ (say over the rationals) imply the existence of a $T$-equivariant homotopy equivalence between $X$ and $Y$?
One set of sufficient conditions may be obtained if your map plays nicely with respect to subspaces fixed by closed subgroups. Let $X$ and $Y$ be $G$-spaces for any $G$ (not only the torus) and assume that you have an equivariant map $f:X \to Y$. If for any closed subgroup $H < G$ the induced map $X^H \to Y^H$ of $H$-stable subspaces induces homotopy equivalence, then in fact $f$ admits an equivariant homotopy-inverse.
You can find all this and much more in John Greenlees and Peter May's lovely notes here.