A fundamental question regarding Chanukah is what the purpose of lighting Chanukah candles is and what כוונות we should have in mind when we light Chanukah candles. The answer is that we are fulfilling Pirsumei Nissa, spreading the miracle of Chanukah, and by doing so we show our belief in Hashem and His miracles. This is accomplished by lighting the candles, either in our windows or outside our homes.
An obvious question comes to mind: what is the point of lighting one’s Chanukiah in public view if the majority of the public is not Jewish? Does the Mitzva of Pirsumei Nissa mean we must spread the news of the miracle to non-Jews as well? If it does not, then we should light inside so that Jewish people inside the house will see the lights.
Rabbi Herschel Reichman, Rosh Yeshiva at Yeshiva University, related that according to the Lubavitcher Rebbe and Rav Moshe Feinstein, zt”l, Pirsumei Nissa does apply to non-Jews. Rav Yosef Dov Soloveitchik is cited as agreeing with this assertion. According to these opinions, a fundamental difference between Chanukah and Purim emerges: We know that the Mitzva of reading the Megila is also Pirsumei Nissa, yet we do not read the Megila in public to enable Jews and non-Jews alike to hear about the miracle of Purim.
To explain this difference we need to recognize that the miracle of Purim was that Jews were saved and elevated above their non-Jewish oppressors. One of the reasons we light candles on Chanukah is that the one-day supply of oil lasted for eight days. This miracle, which happened in the Bait Hamikdash, was not only for the Jewish People, but also for every nation of world. Non-Jews were permitted to bring Korbanot in the Bait Hamikdash, and the seventy bulls brought on Sukkot were brought for the seventy nations. The Megila is read inside, privately, so as not to embarrass the non-Jews or appear as if we are making fun of their defeat, but we go out of our way to put the Chanukiah outside to remind the world of the miracle that occurred in the Bait Hamikdash.