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Is there any practical application to this? I'm not one of those silly people who only cares about math if it can make a better lawnmower or whatever, but I'm just curious as to whether this could make certain algorithms easier or something, or if it's "just" a pure number theory result.

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The conjecture has been verified already for large enough n that it can be applied without a real proof; a proof won’t give any new “real world” applications. Like with Fermat’s last theorem, the real interest in a proof is in the ideas that would be developed along the way. (For example, on the way to proving FLT Wiles established the modularity theorem for a class of elliptic curves, and his work was extended to a proof of the full modularity theorem in 2001.)

Tao mentions in the article that there may be applications in cryptography.

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