The Travelling Salesman problem is a good demonstration of where evolutionary algorithms (EAs) can be useful for finding good solutions to difficult problems. This applet allows you to compare evolutionary and brute force approaches to the Travelling Salesman problem for up to 15 cities. You can tweak the parameters of the evolution to see how it affects results. The applet uses the Watchmaker Framework to implement the EA.
In a quarter of a second we can find a solution to a problem that would take three days to brute force (WARNING: Don’t try to brute force a result for 15 cities unless you have a seriously powerful machine). The EA solution is not guaranteed to be optimal (from observations it seems to be optimal about 90% of the time for this example and is “near”-optimal in the other 10% of cases).