I've started to look into the whole unit testing/test-driven development idea, and the more I think about it, the more it seems to fill a similar role to static type checking. Both techniques provide a compile-time, rapid-response check for certain kinds of errors in your program. However, correct me if I'm wrong, but it seems that a unit test suite with full coverage would test everything static type checking would test, and then some. Or phrased another way, static type checks only go part of the way to "prove" that your program is correct, whereas unit tests will let you "prove" as much as you want (to a certain extent).
So, is there any reason to use a language with static type checking if you're using unit testing as well? A somewhat similar question was asked here, but I'd like to get into more detail. What specific advantages, if any, does static type checking have over unit tests? A few issues like compiler optimizations and intellisense come to mind, but are there other solutions for those problems? Are there other advantages/disadvantages I haven't thought of?