The half-edge data structure, used in computational geometry.
I remember when I first discovered it, it drove home the point that choice of data structure can really matter - it made a bunch of problems I'd been working on much easier to solve, as it gave a different way of looking at things.
PureScript isn't just for business logic at all - it has its own UI libraries, React bindings, DOM API bindings, and so on. It is more often used for front-ends than anything else that I've seen.
You only need to use one of spago or bower (spago being the recommended choice now), and that's because npm style dependencies are not suitable for PS.
As for why you'd choose it over TypeScript... well, you'd do so if you want to use a functional language that doesn't allow arbitrary effects and has a sound type system.
I don't know whether this is true or not, but either way, working with Haskell makes my job more enjoyable than if I was writing some other language anyway.
I remember when I first discovered it, it drove home the point that choice of data structure can really matter - it made a bunch of problems I'd been working on much easier to solve, as it gave a different way of looking at things.