A sinker is bottom tackle. It fits a bottom rod and nothing else, so it belongs to feeder and ledger fishing rather than to float work. Its job is to carry the rig out and hold it on the bottom where the bait can settle and the groundbait around it can do its work.
Weight is the only thing about a sinker that matters. There are five, from 28 g up to 85 g, in three finishes each, but only the 56 g is actually for sale: the other four weights carry no price in any currency and a sell value of zero, the same as the oversized hooks that never made it into the game. So in practice the choice is a finish, at 200 credits, or the Omamori Omiyage souvenir at the same weight.
Where you would have a choice, the reasoning is the ordinary one: enough weight to reach the spot and hold it, and no more than that. A heavier lead casts further and sits tighter against a pull, and a lighter one settles more gently and suits a soft bottom.
Do not read too much into the finishes. Weight is all a sinker has: there is no drag, noise, visibility or sink rate to separate them, and the three finishes behave identically at every weight. They look different and fish the same, which the in-game blurb more or less admits when it describes them as shiny, stealthy, or just rocks with a hole drilled in. Buy whichever you like the look of.
There is no stealth advantage for stone in clear shallow water, and no better hold for iron in a current. Those are true of real sinkers and would be reasonable design choices, but the game does not model them. If a later patch separates them, this page will change.
