What is the practical use of a phase response plot?
Can (will) affect stability.
A filter with enough phase shift is called an oscillator :-) :-(.  
When multiple frequencies are involved the waveform of the output signal will be affected by variations in phase delay of its components. eg to use the obvious example, a square edge becomes rounded as the phase, equivalent to delays, of component fequencies vary.
This effect is important enough that eg the classic Butterworth, Bessel and Chebychev transfer functions have been considered worth dealing with, designing for and talking about for decades. 
Butterworth - maximally flat, phase response of components vary somewhat but amplitude flatness is more important in the related application.
Bessel - maximally constant delay - phase MATTERS even if flatness and absolute cuttoff rates are not well controlled.
Chebychev - maximal amplitude cutoff rate. We don't want no stinkin phase considerations, ie the ability to drop off the edge of the passband  suddenly is most important.


Note that at "any distance" from the filter edge the amplitude response per pole of each type is essentially the same  at6dB/octave, 20 dB/decade.