Imperial portraiture is an intensely studied phenomenon. Nothing about it seems arbitrary - imperial portraits can be dated and distinguished by the number and positioning of the curls along the forehead. For example, the tousled hair of Augustus was likely modeled on the famous portrait of Alexander by Lysippos, implicitly hitching Augustus' reputation to Alexander's horse.

Nero's portraiture gets a bit less attention than Augustus' but here are two suggestions: first, the most likely reason is that beards were a decidedly Greek thing. Philosophers have beards, Pericles had a beard (that's probably the comparison Hadrian was going for with his very full beard). Although our sources on Nero are very biased, there is still reason to think that he styled himself as a more cultured, more Hellenized emperor. Having himself portrayed with a beard encouraged this connection.

Second, slightly less likely, but perhaps more entertaining theory, Nero was trying to hide his weight. The Romans were big believers in physiognomy and just as prone to snap judgements about appearance as we are. Nero's beard starts to appear in imperial portraiture as the emperor starts getting fat. The chinstrap may have simply been a way to disguise the loss of the defined chin that had been part of his earlier portraiture.