Why are functional tests not enough? What do unit tests offer?

I just had a conversation with my lead developer who disagreed that unit tests are all that necessary or important. In his view, functional tests with a high enough code coverage should be enough since any inner refactorings (interface changes, etc.) will not lead to the tests being needed to be rewritten or looked over again.

I tried explaining but didn't get very far, and thought you guys could do better. ;-) So...

What are some good reasons to unit test code that functional tests don't offer? What dangers are there if all you have are functional tests?

Edit #1 Thanks for all the great answers. I wanted to add that by functional tests I don't mean only tests on the entire product, but rather also tests on modules within the product, just not on the low level of a unit test with mocking if necessary, etc. Note also that our functional tests are automatic, and are continuously running, but they just take longer than unit tests (which is one of the big advantages of unit tests).

I like the brick vs. house example. I guess what my lead developer is saying is testing the walls of the house is enough, you don't need to test the individual bricks... :-)

Solution

Off the top of my head

Unit tests are repeatable without effort. Write once, run thousands of times, no human effort required, and much faster feedback than you get from a functional test
Unit tests test small units, so immediately point to the correct "sector" in which the error occurs. Functional tests point out errors, but they can be caused by plenty of modules, even in co-operation.
I'd hardly call an interface change "an inner refactoring". Interface changes tend to break a lot of code, and (in my opinion) force a new test loop rather than none.