I've been writing some programs to deal with the Maya Calendars, where it is an issue.

On some level, it doesn't matter if an inscription corresponds to "August 31, 83 CE" or "September 2, 83 CE" (a date potentially inscribed at the site Takalik Abaj in Proleptic Gregorian or Julian), but it's nice to try to get the details correct. And if you want to correlate with other events in history (especially if you can find astronomical event data from Europe), you need to be at least consistent.

Plus, it's just fun to think about these events from long ago in terms (dates) we understand.