There are a million ways in which a digital audio mixer is different from an analog audio mixer. They handle incoming audio differently, they can send it out differently, they handle distortion and overloading differently, and the list goes on.
Your basic and most fundamental difference between a digital audio mixer and an analog audio mixer is in the added functionality that a digital mixer offers you at the interface. Leaving aside all the arguments as to sound quality and so forth, the cool features of a digital mixer include the fact that a digital mixer is usually all you need to run a successful performance. By this I mean that a digital audio mixer will generally have the following features built right in to the mixer:
So, you can see that you can throw out a whole bunch of outboard processing gear if you just have a digital mixer. Some even have touch screens to make the interface even more easy.