filterbank: understand the different responses at the center frequency of each filter
Normalizing the filterbanks by their widths is optional and totally up to you (similarly to the warping scale Mel/Bark). Depending on your application, you can start without normalization and see what results you are getting. Personally I prefer to keep it fixed and have one knob less for turning. There are more important parameters to tune, such as warping scale, number of banks and coefficients.

For example in the case of DNN's normalization is redundant, since you will be normalizing the input features anyway (at least that's what you should do), so any way you will end up with the same distribution.