I surmise you mean set difference. I.e., P(A) - P(B) = {x |x in P(A) but x not in P(B)}.

In this case, the empty set (0) is NOT in P(A) - P(B).

0 is in P(B) so it cannot be in P(A) - P(B).

That 0 is a subset of every set doesn't entail that 0 is a MEMBER of every set. 0 is a subset of P(A) - P(B) but 0 is not a member of P(A) - P(B).

You're right when you say P(A) - P(B) = {{1}}, since {1} is the only element of P(A) that is not an element of P(B).