The practical uses of a neural network is pretty much everything.
- Recognition / detection in vision
- Artificial intelligence in games
In short : neural network can do pretty much everything as long you're able to get enough data and some efficient machine to get the right parameters. My professor told me once that some competition between schools in neural network turned in the end to the one having the most efficient computers. Because neural network for real problems need a lot of calculation power for the learning phase.
However neural network have some drawbacks, when it come to classification : the value of the parameters of the networks means pretty much nothing to a human.
There is other algorithm that produces results that you can read. I remember of one but only the French name, it's a Lasso regression with an absolute value of a lambda variable. Though it won't ever beat a neural network, you can read the results from the learning and find unnecessary data in order to exclude them. So it can in the end improve the results of your neural network or whatever you use after (SVM, kernels ...).
While neural networks are really powerful, they're not magic and won't beat every other algorithm in all fields. For example, the capacity of learning really fast might make a Support vector machine a better fit, since some can learn and process classification with incredible speed and accuracy.