There are 3 halachot that I have located that are related to the reason of Pirsumei Nissa - publicizing the miracle of Chanukah (Undoubtedly, there must be others:
- Chanukah candles must be lit in a window facing the public (street)
- Candles are lit in shul (See this answer)
- The haftarah for Chanukah is said even if a mistake was made in the reading of the Torah sections on Shabbat Rosh Hodesh Chanukah. (See this answer)
Why is publicizing the miracle specifically relegated to Chanukah?
By comparison, a great miracle happened on Purim, too. Why don't we publicize this miracle?
Pesach was also a great miracle. Yet reading the Hagadah is not called pirsumei nissa, and there's no mitzvah on Pesach to "publicize" it.
What's so special about Chanukah that warrants having to publicize it. And, what exactly are we "publicizing"?