In the last year or so I have learnt the hard way that Integration tests to are not enough.
Throughout this post I'm using the term "integration tests" to refer to a more specific kind of tests which are also known as User Acceptance Tests (UAT)
There is value in writing Unit tests (and specifically doing TDD) that is not gained by writing integration test.
Integration test are design agnostic. The main goal of integration tests is the prove that the system is working correctly, from the USER point of view. At most, when these tests are done TDD style they will help the API's of the system to look better. If we want to leverage tests to drive the system technical design, the tests must be aware of the different parts which construct the system design (Units) and explicitly test each of them (Unit tests).
Software testing is an exponential problem by nature. As a system gains complexity the amount of test cases needed to cover all possible path and behaviors, grows exponentially. Therefore its not really possible to cover everything by integration tests. A Possible feasible approach is "Divide and Conquer". The system is cut into several parts (units), and each part is tested separately (unit tests). To those test we add specific tests that will make sure that all the parts play well together (integration tests)
Software systems are complex by nature. A large part of a system complexity is actually caused by the need to handle all things that can go wrong. I think I can fairly state that in most cases the number of tests written aimed to make sure the system doesn't break in extreme cases is larger then the number of tests covering normal system behavior. The thing is, that when thinking as a user, its really hard to look at the system as a whole and figure all the things that can go wrong. It's hard enough for us to think what the user actually needs, so trying at the same time to look at the system and find all the corner cases is really hard. It's generally a lot easier (for most developers) to look at each part of the system (units) on its own, analyze all possible failures and write tests (unit tests) that make sure this failures are handled properly.
To share my past experience, I’ve fallen into the Mistake of assuming that integration tests will be enough. After neglecting for enough time testing at the unit level, I observed an increase in our incoming defect rate. and the most annoying fact in all those incoming defects were that most of them would have been caught by proper testing at the unit level.