In pointless topology and point-set topology, there are some theorems in pointless topology that can be proven without the axiom of choice, but have corresponding theorems in point-set topology that do require the axiom of choice.
What does this mean? Here's what I think it means:
1. First of all, we have some ideal form of space that we want to prove.
2. If the framework is good enough (Pointless topology captures it), we're done.
3. If the framework is not (Point-set topology doesn't) we add in some axioms that do capture that ideal form (axiom of choice). And then we prove it.
So it's not the axioms or the formal framework that's important (this is formalism) but rather the ideal form we perceive (back to Platonism). Platonism got a bad reputation because so many crank philosophers glorified euclidean geometry as divine revelation... but I find it impossible to make sense of mathematical progress without resorting either to Platonism or some other divine mind responsible for capturing the forms.
Many people do not know this, but there was a movement in mathematics akin to materialism in physics called formalism, which was an attempt to take God out of mathematics. However, it is a dead movement now and I suspect as many mathematicians grow restless a return to Platonism will happen.