When it is likely that the response follows a distribution that is a member of the exponential family, a generalized linear model may be more appropriate to use, and moreover, when we wish not to force a parametric model onto our data, a non-parametric regression approach can be useful. The importance of being able to model the variance as a function of the mean lies in improved inference (in a parametric setting), and estimation of the regression function in general, for any setting.
