Is difference of two projection matrices positive semi-definite or negative definite or indefinite?

Presumably the matrices are real. The difference in question is $RR^+-QQ^+$, which is a difference of two orthogonal projections. Therefore begin{align} &RR^+-QQ^+preceq0\ Leftrightarrow &operatorname{range}(RR^+)subseteqoperatorname{range}(QQ^+)\ Leftrightarrow &operatorname{range}(R)subseteqoperatorname{range}(Q)\ Leftrightarrow &W^{-1/2}operatorname{range}(S)subseteq W^{1/2}operatorname{range}(S)\ Leftrightarrow &W^{-1/2}operatorname{range}(S)=W^{1/2}operatorname{range}(S) text{(because both sides have equal dimensions)}tag{1}\ Leftrightarrow &operatorname{range}(S)=Woperatorname{range}(S).tag{2} end{align} However, by a similar argument, we also have $RR^+-QQ^+succeq0$ if and only if $(1)$ holds. Hence $RR^+-QQ^+$ is either both positive semidefinite and negative semidefinite, or indefinite. In other words (and by $(2)$), $RR^+-QQ^+$ is zero when $operatorname{range}(S)$ is an invariant subspace of $W$, or indefinite otherwise.

If the result is positive semi-definite for some $$W=W_1succ0,$$ then it will be negative semi-definite for $$W=W_1^{-1} succ 0,$$ since $R$ and $Q$ will simply switch places.