Here is a link to the most straightforward explanation around these concepts I have seen.

In case you don’t want to drill down to the link above, following is an excerpt.

I’ve tried answering this question so many times and in so many ways: “What’s the difference between a use case and a user story?”

I know what each is, I know how to describe each; I’ve described each and even how to relate the two. I’ve finally concluded that a user story is to a use case as a gazelle is to a gazebo.

See if you can describe the difference without just enumerating what each is.

As the Mad Hatter asked: “Why is a raven like a writing desk?”

A better answer is A user story is the title of one scenario whereas a use case is the contents of multiple scenarios.