What is the difference between phase delay and group delay?
First of all the definitions are different:


Phase delay: (the negative of) Phase divided by frequency
Group delay: (the negative of) First derivative of phase vs frequency 


In words that means:


Phase delay: Phase angle at this point in frequency
Group delay: Rate of change of the phase around this point in frequency. 


When to use one or the other really depends on your application. The classical application for group delay is modulated sine waves, for example AM radio. The time that it takes for the modulation signal to get through the system is given by the group delay not by the phase delay. Another audio example could be a kick drum: This is mostly a modulated sine wave so if you want to determine how much the kick drum will be delayed (and potentially smeared out in time) the group delay is the way to look at it.