The following is based on my limited understanding of a machshavah shiur from Rav Moshe Shapira zt”l for parshas Mikeitz 5768. It's part of a devar Torah I wrote up in 5779.
Hashem told Moshe: “…I will give you the stone tablets, and the Torah and the Mitzvah…”1. The stone tablets contained the Ten Commandments. The Torah is the Torah Scroll which we have today2. What does “the Mitzvah” refer to? We are taught that it refers to the explanation of the Torah3. How does the word “Mitzvah”4, which is usually translated as “commandment”, connote an explanation? What is an explanation? It’s a revelation of the true intent or meaning of something. So too a Mitzvah. Mitzvos reveal the Divine Will that is laden within the Torah5. In that sense, a Mitzvah is an explanation of the Torah.
We are taught6 that when Malachi, the last prophet, died, Alexander began his conquest7. A main difference between the time of the prophets and afterwards is the ability to describe something’s essence. A prophet is able to relate something, whereas the Greeks and those that followed them can only relate about something. The Greeks and the scientists are great at providing descriptions of something, but that they can never discuss its essence. That requires Divine knowledge, of which only the prophets have. Since the story of Chanukah occurred after the cessation of prophecy, we can no longer describe the essence of the miracle that took place. Any public recitation or communal storytelling wouldn’t do the miracle justice.
However, reciting from the Torah or the Megillah is different, as they were written by prophets. Miracles that are described in those books aren’t just descriptions about the miracle, they convey the essence of the miracle. We are then able to publicize the miracle in the utmost way. So too when we perform a mitzvah. Since a mitzvah is a revelation of the Divine Will, it’s an explanation of the Torah, by lighting the menorah we are publicizing the miracle8. We aren’t simply describing it, we are revealing its essence9.
I admit this doesn't address why we read the Megillah in shul and not in public. I hope however that this is the start of an answer.
1 Exodus 24:12
2 Berachos 5a
3 Rambam’s *Hakdamah *to Mishneh Torah, based on Berachos loc. cit. Cf. Rashi ad. loc. who seems to explain that it refers to the 613 mitzvos
4 Rav Moshe Shapira pointed out that the word מצוה is a very strange word. It sounds tangential but I believe it was essential for his presentation. Unfortunately, I only understood it enough to leave it as a footnote. What he says is really, the word should be ציווי, which means commandment. Why is the word referred to instead as “Mitzvah”? There are certain roots which can take on different forms, and one form is a combination of them all. For example, the word שופט is a judge, and the נשפט is the one who was judged, the defendant, and the שפיטה is the judgement. משפט is usually the word which ties them all together. It represents the judge ruling a judgement towards or against the defendant. We find a similar case with a loan. A מַלוה is the lender, the לוה is the borrower, and the הלוואה is the loan. All together they can be summarized with the word מִלוה, which connotes a lender giving a loan to the borrower. So too the word Mitzvah. There’s the מְצוֶה, the commander, who commands the מְצוּוֶה, the commanded, the ציווי, the commandment. All of this is summarized with the word מצוה
5 The meforshim (such as Tzafenas Paneach Chatimah to Seder Zeraim, Tikkunei Zohar p. 73a, brought by the Shela Yoma Chapter Derech Chaim Tochechas Mussar 16) point out that if you take the first two letters of מצוה, and perform the א"ת ב"ש conversion, you get י and ה, which are the first two letters of Hashem’s name, combined with the last two letters of מצוה to get the last two letters of Hashem’s name. This name is known as the שם המפורש, literally the explained name. The Rosh (Yoma 8:19) says that this name includes every other name of Hashem. Really, the entire Torah is made up of Hashem’s names (Ramban’s Introduction to Chumash)
6 Rav Moshe Shapira got this from Seder Olam Rabbah Chapter 30
7 It is also when the yetzer hara for idol worship ceased, leaving only a yetzer hara for heresy and lust, which were well-abused by the Greeks
8 Even though the mitzvah of lighting the Menorah was enacted after the cessation of prophecy, we can still say that it reflects the Divine Will. I’d say that this is so because the concept of Chanukah is alluded to in the Torah (see Ramban to Numbers 8:1). However, I believe Rav Moshe Shapira addressed this directly. Seder Olam Rabbah loc. cit. says that once there was a cessation to prophecy, go and listen to the Sages. Rav Moshe Shapira explained that this is because they still have some element of ruach hakodesh. Since the Sages were the ones to institute Chanukah, their ruach hakodesh can still be accessed through performing the mitzvah of lighting the menorah
9 Rav Moshe Shapira added that the same is so with the mitzvos of Pesach. Eating matzah and marror not only are fulfilling Hashem’s commands, they reveal the essence of the miracles of Pesach