What different uses do we assign to salted butter versus 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.
The short answer is that unsalted butter is for cooking with, salted butter is for spreading on things (biscuits, pancakes, etc.). Salted butters vary in the amount of salt they contain, so when cooking you should use unsalted butter and control the salt level yourself.
I believe I've heard that in some parts of Europe salted butter isn't even sold. If you want to butter your croissant, you use unsalted butter and sprinkle a touch of salt for flavor.
In the UK most butter is salted and just labeled butter. Most people use this for everything, unless a recipe calls for unsalted butter particularly. I assume our baking recipes take that into account, but lots of our baked goods, eg shortbread, wouldn't taste right without it.
Well, you can assign whatever uses you want, of course. But in general, unsalted butter is used anywhere that you can't or shouldn't taste for saltiness, or where you want to avoid the salt, and salted for anything that you prefer it on, or where there's no worry about getting too much salt without knowing it. This is because the saltiness varies so much with salted butter, so you never know how much salt you're getting.
Of course, lots of people use unsalted butter for most everything--many prefer it on fresh bread, for example--and there are plenty of people who don't worry about using salted butter for everything either.