Recently we have been adding automated tests to our existing java applications.
What we have
The majority of these tests are integration tes
The question has a philisophical part for sure, but also points to pragmatic considerations.
Test driven design used as the means to become a better developer has its merits, but it is not required for that. Many a good programmer exists who never wrote a unit test. The best reason for unit tests is the power they give you when refactoring, especially when many people are changing the source at the same time. Spotting bugs on checkin is also a huge time-saver for a project (consider moving to a CI model and build on checkin instead of nightly). So if you write a unit test, either before or after you written the code it tests, you are sure at that moment about the new code you've written. It is what can happen to that code later that the unit test ensures against - and that can be significant. Unit tests can stop bugs before tehy get to QA, thereby speeding up your projects.
Integration tests stress the interfaces between elements in your stack, if done correctly. In my experience, integration is the most unpredictable part of a project. Getting individual pieces to work tends not to be that hard, but putting everything together can be very difficult because of the types of bugs that can emerge at this step. In many cases, projects are late because of what happens in integration. Some of the errors encountered in this step are found in interfaces that have been broken by some change made on one side that was not communicated to the other side. Another source of integration errors are in configurations discovered in dev but forgotten by the time the app goes to QA. Integration tests can help reduce both types dramatically.
The importance of each test type can be debated, but what will be of most importance to you is the application of either type to your particular situation. Is the app in question being developed by a small group of people or many different groups? Do you have one repository for everything, or many repos each for a particular component of the app? If you have the latter, then you will have challenges with inter compatability of different versions of different components.
Each test type is designed to expose the problems of different levels of integration in the development phase to save time. Unit tests drive the integration of the output many developers operating on one repository. Integration tests (poorly named) drive the integration of components in the stack - components often written by separate teams. The class of problems exposed by integration tests are typically more time-consuming to fix.
So pragmatically, it really boils down to where you most need speed in your own org/process.