I agree both approaches are important, and it's totally ok if they overlap. If your unit tests have some overlap on your integration tests, that's nbd especially seeing as you can run your unit tests in parallel.
EDIT: actually I'll make a much bolder claim: even if your unit tests are making flawed assumptions about the underlying dependencies, it's still pretty much fine so long as you also exercise those dependencies in integration tests. That is, even somewhat bit-rotted unit tests with flawed mocks and assertions are still valuable because they exercise the code. More shots on goal is a great thing even if they're not 100% reliable.