Suppose $M, N$ are self-adjoint operators, let $E_M$ be the spectral projection measure of $M$ and $E_N$ be the spectral projection measure of $N$.
If we denote $E_{M-N}$ by the spectral projection of $M-N$, does the following relation hold?
$E_{M-N}=E_M-E_N$
It's actually kind of hard to find any example where the equality holds. The equality does hold when $M,N$ are both projections with $N\leq M$. I cannot imagine another situation where it holds.
If your equality held, you immediately get that $E_N\leq E_M$ (since projections are positive). So $E_N$ is a subprojection of $E_M$. This implies that $\sigma(N)\subset\sigma(M)$. That's far from enough, though.
Even when the operators commute, it is easy to find counterexamples. For instance let $$ M=\begin{bmatrix} 1&0\\0&0\end{bmatrix},\qquad N=\begin{bmatrix} 0&0\\0&1\end{bmatrix}. $$ Then $$ E_M(\Delta)=M\,\delta_{1}(\Delta),\qquad E_N(\Delta)=N\,\delta_1(\Delta), $$ so $$ E_M(\Delta)-E_N(\Delta)=(M-N)\,\delta_1(\Delta),$$ which is not positive (nor projection-valued).