Forgetting the gigantic advantages it gives you over mere bugs, I'm just as impressed at how it helps you resolve higher level issues.
For example, recently I moved a project I was working on months ago from one machine to another. Tests had lots of failures, despite no changes. Turns out I had an unpatched library installed, a few libraries of the wrong version, a slightly different database version, stuff like that. My tests picked all that up and I could vendorize stuff or tighten up my dependencies. Without the tests, I'd have been scratching my head for ages figuring out "what" wasn't working.