I think an example is this: for a concrete set of data, see the distributions on page 28 of this document (page 40 of the PDF) http://mi.eng.cam.ac.uk/reports/svr-ftp/auto-pdf/harpur_thes...

These different distributions are difficult to distinguish with statistics. However, if you can see the shape snd therefore know the "rule" or the structure of the distribution, it's easy to design/train a network to recognise them.

(note that the content of that PDF is not about this problem per se)