The difference between those are that unit tests one object and one object only. Typically mocking anything else outside the SUT (System under test).
Integration tests a system of collaborators, or multiple objects that work in concert. Functional tests a system from the outside into the centre as a full stack test. User stories test interactions and outcomes.
The difference between functional and user stories is that a functional test exercises a system with a focus of "given this kind of input, I expect this output back" and user story is more focused on "When I perform this kind of thing, given the system is in this kind of state or has this data, I expect to see this kind of thing as a result."
Functional can also test the side-effects rather than just the input and output.
Credit to @Spaceghost for the above.