The biggest difference of agile methods from traditional waterfall is the short feedback loop. The whole concept of agility in essence is no more, than "build the most important piece, evaluate, adjust, and repeat". Automated tests in the agile methods serve as a very important tool for shortening the feedback loop.
In the most of traditional processes automated tests are mimicking the manual test procedures. The tests are often written not by the code developers. Tests usually involve testing the functionality of the whole system. Test results come to the original developers quite late and tests don't take into account the latest code changes. Automating functional and system tests is a good thing. It makes testing a little easier, more effective and saves some money for the company. However, it is no more, than automating a small piece of the manual labor.
In agile methods the majority of the automated tests consists of unit tests that verify the smallest possible pieces of software and can be executed very quickly. It makes it possible to execute the test set many times a day or even many times an hour and shortens the feedback loop even more.