I believe that both will remain rare in audio drama, though binaural systems may become important in virtual reality. Therefore, I don't want to dwell too long on either. True quad captures and plays back sound on four channels. The listener is in the middle. Theoretically, the action can take place anywhere around the listener.
True quad works only for non-broadcast audio.
Broadcast quad is matrixed, not true - that is, there are only three channels, two in front of the listener and one in back. The third and fourth speakers on the listener's receiver play back the same signal, doctored to sound like two discrete channels. Matrixed quad renders the two back channels useless except for environmental overtones. Binaural captures sound on two channels in a way that imitates human ears. The mikes react to sound resonating in the shell like ears reacting to sound resonating through the skull.