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But if you assume rook contiguity then it seems equivalent to hexagons. All neighbors share a full edge, all neighboring centers are equidistant.

I get that you're saying hexes are almost always a better representation. I still don't see a concrete example of why, for geographical indexing specifically.

[Edit: sibling reply explained that at the end of the day, it's not about indexing but rather route planning.]

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Let's say you are looking for the closest gas station. One that is in the close corner of a "diagonal neighbour" would be closer than most points in the "edge neighbours". So if you want to find something nearby, you'd usually want to look at all 8 neighbours. The hexagonal neighbours look more like a circle centered in the original hexagon, thus more convenient for that purpose.

and if you assume cows are spherical and frictionless it's quite convenient

sometimes roads go diagonal

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