The difference is that it's very easy to construct a non-Euclidean geometry in the physical world, at home, on your table, but it's impossible to construct an object anywhere in the real Universe where the Uncountable Axiom of Choice applies. It's purely a mathematical game of pretend.

They are as far apart from each other as possible, as similar as polar opposites.

Banach-Tarski is resolved by deciding that "the real numbers" aren't real, and nothing is lost except for dubious overly-simple proofs.