Because applications have state. In fact, the vast majority of reads in any applications are stateful (the database backing your app is also state, you know; I'm not talking just about session state here).

By "pure" here I mean that we're talking about a pure function of its inputs.

Why do you need to distinguish between those? In fact, if the function is pure, why not run it on the client?

Because the server has more computational resources, for example. It's a very rare scenario in general, but it does happen.

Why distinguish? Because if you know that the results are not affected by server state, they can be cached much more aggressively.

You mean like a cursor in a database?

Logging/analytics? Paywalls?

Rate limiting APIs?