If it was just about asserting correctness and detecting bugs, a round of manual testing would do just as well. You might happen to catch some bugs by running your automated tests, but you won’t catch what you didn’t design the tests for. Simplifying, you cannot reveal a bug that you don’t know of, and if you knew about it you could just fix it.
The value of automated tests is that, once written, they become part of your test suite, pinning down the invariants that should remain even when the implementation details are changed. Every time someone changes some code, the test suite informs them if some of those invariant was inadvertently broken.
The biggest cost of writing tests is payed only once, but you can run the same test as many time as you want during the development process. Therefore, they are efficient to safeguard against breaking functionality while evolving your project.
Inspections and manual testing have a cost every time they are performed. They are a better way to assert correctness and quality of newly added features, but re-inspecting or manually testing everything upon every change would be inefficient.
