Why do we care about the multiplicity of poles and zeros in rational Z-transforms? In the DSP class I'm taking, a lot of the questions ask me to list the multiplicity of a pole or zero in a rational Z-transform. For example, the multiplicity of the zero at $z=1$ in :$\frac{(z-1)^2}{(z-5)}$ is $2$.

What knowledge do we gain by knowing the multiplicity of some pole or zero? 

Of course, if we know the multiplicity of all the poles and zeros, we can describe the z-transform of a rational function, but this is not a very satisfying answer.
Multiple poles have amplitude and phase profile different to that of single poles.