In Kabbalah it is taught that the decrees of Rosh Hashanah are not fulfilled until the end of Chanukah. There are hints to this in the Chumash as well.

Parshas Vayeshev is always read in close proximity to Chanukah. There we read the story of Yehuda and Tamar. The scene in heaven is played out this way: Yehuda represents Hashem and Tamar represents Klal Yisroel. The Satan tells Hashem that Tamar/Klal Yisroel sinned, so Yehuda/Hashem decrees we must be punished. Tamar then says, “To the one who owns the seal, cord and staff – that’s the one who is guilty. Pardon me on account of these 3 things.” Klal Yisroel also asks Hashem to pardon us on the account of those 3 thing: the seal – we went the extra mile to find oil with a seal; the cord – the wicks for the Chanukah candles, and the staff – the walking stick representing our difficult path in life. Yehuda/Hashem says, “She is right,” and exonerates Tamar/Klal Yisroel.

There is another hint to this concept in Parshas Noach. It says in Bereishis (8;6), “Vayehi mikeitz arbaim yom…” (And it was at the end of 40 days). This can refer to the 40 days from Rosh Chodesh Elul to Yom Kippur. If one hasn’t done proper teshuva yet and thinks it’s all over… “Vayiftach Noach es Chalon” can be interpreted to mean “Hashem opens a window of opportunity to do teshuva on Noach” [Nun-Ches = Neiros Chanukah] When the word Chalon is broken down into two parts: Ches-Nun = Chanukah Neiros and Lamed-Vov = 36 candles, the equivalent of all the candles lit on Chanukah.

Did You Know:
In the Diaspora, on Chanukah we spin a dreidel that has 4 Hebrew letters on it: Nun Gimmel Hey and Shin (which stands for Nes Gadol Hayah Sham; A great miracle happened there). The gematria (numerical value) of Nun-Gimmel-Hey-Shin is the same as that of the word Moshiach (358).

In Israel, the dreidel has the letters Nun Gimmel Hey and Pey (which stands for Nes Gadol Hayah Poh; A great miracle happened here) The gematria of Nun-Gimmel-Hey-Pey is the same as that of the word Menachem (138). Menachem is one of the names for Moshiach.
– Maharsha”h Pesachim, Daf Hey

Q. Did you know that many Jews don’t say the blessing over the Chanukah candles “L’hadlik ner shel Chanukah,” rather they say “L’hadlik ner Chanukah“?
A. This is because the candles of Chanukah are different from Shabbos candles (where all say “L’hadlik ner shel Shabbos“), since one cannot benefit from the light of the Chanukah candles. Therefore, the candle is not a candle for Chanukah (ner shel Chanukah), it is a Chanukah candle (ner Chanukah).

How are Chanukah and Purim different?
On Chanukah the nes (miracle) was an open and revealed miracle from above (l’maalah) while on Purim the nes was hidden and seemingly natural (l’matah). To commemorate the way the miracles happened, on Chanukah we spin the dreidel from the top, while on Purim we spin the gragger from the bottom.