Proof that $A^* - \{x\} = A^+$ such that $A=\{a,b\}$ and $x \in \{a,b\}^*$
Your problem is that in going from $A^*-\{x\}$ to $\{a,b\}^*-\{\{a, b\}^*\}$ you went a step too far. In the first expression, $x$ is a single string over the alphabet. Replacing that with the general expression $\{\{a,b\}^*\}$ is incorrect, if for no other reason that $\{\{a,b\}^*\}$ is a set consisting of a set of words and not a single word.

To complete this answer, see newbie's comment on your original post.