What different uses do we assign to salted butter vs. unsalted butter?
Salted butter is good only for spreading on toast. In all other cooking applications, without exception, you want to use unsalted butter. For one, the addition of salt in some contexts (particularly baking) will affect chemical reactions, so you want to control how much salt is being used. For another, and more importantly in most applications, using salted butter means you will have less control over actual seasoning.

We keep both salted and unsalted butter at home. Salted, as I noted, is for toast only really.