Uncountable minus countable set is uncountable

Maybe this is a way to see it. You can make it more precise yourself. Assume that $A\setminus B$ is countable. $B$ is countable, so that would mean that $(A \setminus B) \cup B$ is countable (finite union of countable sets is clearly countable). But then $A \subseteq (A\setminus B)\cup B$, so $A$ is contained in a countable set, so it must itself be countable.

First, the two cases you mention don't include all possibilities. For your question, note that the union of two countable sets is again countable.

Suppose that $A-B=A-A\cap B$ is countable. Then you have a bijection between $A-B$ and the set of even natural numbers. Since $B$ is countable then $A\cap B$ is also countable and you have a bijection between $A\cap B$ and the set of odd natural numbers. Now, taking the union of those bijections, you get the bijection between the set $A=(A-B)\cup (A\cap B)$ and the set of natural numbers, which is an absurd because $A$ is uncountable. Thus $A-B$ is uncountable. Q.E.D.

If $A\cap B$ is just finite countable then the proof goes similarly with obvious changes, see comments.