Jeremy, Alan's explanation is perfect. To expand a little further, the regular styles on any arranger keyboard actually make no sounds whatsoever. They are made up of MIDI instructions and pointers that trigger the sounds of the keyboard's tone generator. Here's a diagram (as best as possible with the forum tools):
Audio style - Keyboard note is pressed-->an audio sound at a certain pitch, volume, and limited tempo is triggered and sent to the speakers.
MIDI - Keyboard note is pressed-->a digital code is sent with information about pitch, volume, velocity, unlimited tempo, and loads of other digital info-->tone generator-->speakers.
That's the difference, in a nutshell. The huge advantage of MIDI styles lies in the ability to edit them. For example, if you have a drum kit in an Audio style, it is very difficult to turn down the kick drum without affecting lots of other sounds. In a MIDI style of the same nature, you have access to the digital code that provides the kick drum sound. Hence, you can turn it down or even off. You can also change what kind of instrument is "kicked" if you want. Not possible in an audio style.
Hope that clears it up.