Q. Why is it that we burn the leftover oil, half burned wicks and candles from Chanuka, but we don’t do so for the remains of Shabbath or Yom Tov candles?

A. Horav Shlomo Miller Shlit’a pointed to Tosafos Shabbos (44a) who asks this very question and as explained in Kehilas Yaakov (18) it is mainly due to the fact that the lights of Shabbos are for the purpose of one’s private use, and would expect to use any leftovers. therefore our sages did not impose a issur hana’ah or benefit prohibition on them. Chanuka lights however are mainly lit for pirsumei nissa , or for publicizing the great miracle occurred, not for one’s use and one cannot derive any benefit from the remains, therefore, whatever is left is burned. (See Tosafos Harosh ibid. for two similar answers)

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a