What is so great about the proof of the Poincaré conjecture?

Jordan Ellenberg has a truly wonderful article in Slate.

The entities we study in science fall into two categories: those which can be classified in a way a human can understand, and those which are unclassifiably wild. Numbers are in the first class—you would agree that although you cannot list all the whole numbers, you have a good sense of what numbers are out there. Platonic solids are another good example. There are just five: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. End of story—you know them all. [...]

In the second class are things like networks (in mathematical lingo, graphs) and beetles. There doesn't appear to be any nice, orderly structure on the set of all beetles, and we've got no way to predict what kinds of novel species will turn up. All we can do is observe some features that most beetles seem to share, most of the time. But there's no periodic table of beetles, and there probably couldn't be.

Mathematicians are much happier when a mathematical subject turns out to be of the first, more structured, type. We are much sadder when a subject turns out to be a variegated mass of beetles. [...]

[...] [Perelman's proof of the conjecture of Poincaré] means ... that we can think about proving general statements about three-dimensional geometry in a way that we can't hope to about beetles or graphs.

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Cross-posted at nanopolitan 2.0.