Rama (695:2) quotes from the Maharil (though I don't see it there):
וחייב במשתה ושמחה קצת בב׳ ימים י״ד וט״ו
and one must have a drinking party and joy, a little, on two days: the fourteenth and the fifteenth [of Adar]
(We also don't say tachanun or, I'm pretty sure, eulogies.)
Recall that the celebration of Purim is to commemorate a day of respite after battle. Jews in Shushan fought the 13h and 14th and had respite the 15th. Jews elsewhere fought the 13th alone and had respite the 14th. So for Jews in Shushan to celebrate on the 14th seems incongruous (though perhaps they should fast that day, as they do the 13th). And for Jews in New York (for example) to celebrate on the 15th seems irrelevant: the 15th is just another day on the calendar, no more celebratory than the 16th or the 17th.
The only reason I can think of to celebrate on both days is that Jews elsewhere are doing so, and we like to share one another's joy. But that's conjecture, and seems IMO weak as an obligatory reason. I seek a source for a reason we celebrate both the 14th and the 15th of Adar.
 Indeed, I've heard that given, with no source AFAIR, as the reason we do hakafos in chutz laaretz on lel Sh'mini Atzeres, and in Israel on motzae Sh'mini Atzeres.