Why do we care about perfect sets? When I first heard the name `perfect' set, I thought that these would be especially nice sets that show up a lot. Maybe it's because I haven't done that much analysis, but it seems like perfect sets are introduced, you can prove a funny theorem about uncountability of them in certain spaces, and then you're done and they never come up again.
Why does anyone care about perfect sets? What's so special about them?
 A: One can show that perfect subsets in separable metric spaces have size continuum (i.e. $|\mathbb{R}|$, also denoted $2^{\aleph_0}$ or $\mathfrak{c}$) and that every closed subset of a complete separable metric space (also called a Polish space for short, in honour of the many Polish mathematicians that worked on this theory, a mix of set theory and (metric) topology, called descriptive set theory) is either countable or the union of a countable set and a non-empty perfect set, so has size $\aleph_0$ or $2^{\aleph_0}$. That size dichotomy also holds for all Borel sets (the smallest $\sigma$-algebra on a Polish space that contains all open sets) and this also is proved using perfect sets : uncountable Borel sets of Polish contain homeomorphic copies of the Cantor set, a quite ubiquitous perfect space/set. This can in fact be extended to so-called analytic sets too. Very many of the sets mathematicians use in practice, e.g. in proofs etc are Borel sets or analytic sets and so (using perfect sets, indirectly), the continuum hypothesis holds for such sets. This sort of "explains" why that hypothesis seems so plausible to many mathematicians (while being, it turns out, undecidable by the usual set theory axioms).
So perfect sets, especially Cantor sets, are an important idea in some branches of topology/set theory, like descriptive set theory. But one doesn't meet it very often, unless one studies that field in more depth. AFAIK it's not used much in applied branches (like differential equations, or physics). It's nice to know, it has a nice theory and has beautiful properties (the Cantor space is one of my favourites too). "perfect" is more of a homage to that I think. The word perfect is used elsewhere in topology with another meaning (perfectly normal..), so beware. There are also "perfect maps", closely related to compactness. 
