Tests show how pieces work together outside of the small confines of a live system. If you don't write tests, you probably aren't considering all the ways you component can work outside of the narrow confines when you, as the author, wrote and tested it. And, you probably can't consider how another developer will use your component and change the environmental code around it.
Storybook gives you all the benefits of tests in that you can pull out pieces and play them back in isolation. I see all the benefits of tests (except that I don't see the results in CI) when using storybook and refactoring. Broken stories tell me a lot about code problems.
Storybook is UI testing done right, we just needed to brand it as a story rather than a test. ;)
Congratulations on a big milestone. This is an incredible piece of software that I use on all my projects.