The main difference between these newer networks (besides much improved performance) is that the algorithms can handle "deeper" networks better (more hidden units). If we're talking about Deep Belief Networks, they're not much like the old ANNs. DBNs are generative probabilistic graphical models using Bayesian inference.
Conceptually, going deeper (LOL) allows the networks to learn higher level concepts. For example, a 1 layer ANN (perceptron) can only learn linear functions, while a deep network is able to internally form a belief of what, say, a cat is.
More technically: Much of the work in ML is deciding what your inputs (features) should be. When classifying text documents, should you use word counts, bag of words, word stemming, character counts, etc. Should the model be linear, polynomial, gaussian, trigometric, etc. Deep learners try to automatically do feature selection and control the degrees of freedom in the model for you.
Also, deep learning is catching on in some industries. It has recently had huge successes in speech recognition, and all major companies developing this technology have started using it (e.g. Siri for one).