If $x \in A$ why does it not follow that $x \in A-B$? I understand that if $x \in A-B$ then $x \in A$ because $x \notin B$ but why doesn't the reverse hold true?

Because $A\setminus B$ only includes elements in $A$ which are not also in $B$. Put differently, $A$ can be partitioned as
$$
A = (A\setminus B )\cup (A\cap B)
$$
and if $x\in A\cap B$, then $x\in A$ but $x\notin A\setminus B$. 
For instance, take $A=B=\{1\}$ for a trivial counterexample. Then $A\setminus B = \emptyset$: so $1\in A$, but $1\notin A\setminus B$.
