Cantor spaces occur abundantly in real analysis.
(To see this, note that in such a space, any non-empty perfect set contains two disjoint non-empty perfect subsets of arbitrarily small diameter, and so one can imitate the construction of the usual Cantor set.)
A topological characterization of Cantor spaces is given by Brouwer's theorem:[1] Brouwer's theorem — Any two non-empty compact Hausdorff spaces without isolated points and having countable bases consisting of clopen sets are homeomorphic to each other.
The topological property of having a base consisting of clopen sets is sometimes known as "zero-dimensionality".
But many properties of Cantor spaces can be established using 2ω, because its construction as a product makes it amenable to analysis.