Subtracting empty set from another

You are not removing the empty set. You are removing a set whose only element is the empty set, as $$\emptyset\ne\{\emptyset\}$$

So, $$A\setminus\{\emptyset\}=\{\{\emptyset\}\}$$

Your claim would be true if the set being removed were indeed empty, i.e., $$A\setminus\emptyset=A$$

Put $\{\varnothing\}=a$ and $\varnothing=b$. Hence, $$ A=\{b,a\}\setminus\{b\}=\{a\}=\{\{\varnothing\}\} $$ $$ B=\{b,a\}\setminus\{a\}=\{b\}=\{\varnothing\} $$

It is true that subtracting the empty set gives you back the set you started with: $$ A - \emptyset = A. $$

But $\{ \emptyset \}$ is not the empty set; it has one element, namely, $\emptyset$.