It's not either/or, but integration is more important. As in, if you're going to defer one thing (which happens often), it should be unit tests.

I work for a huge corp, but the startup rules still apply in most situations because we're doing internal stuff where velocity really matters, and the number of small moving parts make unit tests not very useful.

Unfortunately, integration testing is painful and hardly done here because they keep inventing new bad frameworks for it, sticking more reasonable approaches behind red tape, or raising the bar for unit test coverage. If there were director-level visibility for integration test coverage, would be very different.