Current speech recognizers are basically layer upon layer of tricks discovered by researchers over the course of decades. Chop up the input signal. Then take a Fourier transform. Take the log to even the signal out. Do another transform to de-correlate different components of the audio. Add noise to the input. Project down to a subspace. Switch objective functions halfway through training to trade off different kinds of errors. Use more Guassians here. Use fewer there. Pump it into a language model.
It works, and it's a marvel of engineering, but it's not "artificial intelligence." It's pretty much a big stack of statistical hacks piled up over the years.
The nice thing is that a deep belief network can figure out a lot of this structure automatically, much closer to how the brain works.
This paper is actually incremental, not a "leap forward." They've basically replaced two of the middle layers of a speech recognizer (the Gaussian mixture model and hidden Markov model) with a modified neural network. But the exciting thing is that the neural network can start there, and slowly eat its way toward the outer layers, replacing a big stack of hacks with one simple algorithm.