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I find these network visualizations nice to look at, but not all that insightful. They're generally hard to read and track relations outside of the main clusters. Am I missing something?

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No I don't think so. A lot of people call these things "hairballs," and probably a more useful interface would be some kind of faceted browser that allows you to do pivots and look at aggregate stats of the various lenses you can put on top of a graph. Additionally, measurements such as node separation, "betweenness," or perhaps even looking at common chain patterns are probably more useful ways of trying to dissect graph structures.

Let's say you compute how distant/similar two concepts are. Then what? You update a link on Wikipedia?

Yeah I don't know! :D I guess I was talking about graph processing in general: https://en.wikipedia.org/wiki/Betweenness_centrality