There are no major or significant differences between the tradition of the Sephardic and Ashkenazi Jews regarding the customs related to Hanukka. There are only a few minor variations.
According to the Ashkenazi tradition we should say the Berakha: “lehadliq ner SHEL Hanukka,” (….to light the candle of Hanukka). While according to the Sephardic tradition,following the indications of the Shulhan ‘Arukh we should say “lehadliq ner Hanukka,” omitting the word “SHEL”. Interestingly, there is no grammatical or semantic difference between these two versions: one cannot say that one version is right and the other one is wrong. Actually the original version of this berakha (according to Maimonides, MT Hanukka 3: 4) is “lehadliq ner SHEL Hanukka” , similar to “leahdliq ner SHEL Shabbat”. The Sephardic tradition, and the Shulhan ‘Aruch who usually follows Maimonides, is based in this exceptional case in the opinion of the Mequbalim (Rabbis and scholars who follow the Qabbala, like the Ari haQadosh. Incidentally, this is also the reason why Chabad Hassidim also recite the version “ner Hanukka” and not “ner SHEL Hanukka). Some Sephardim, for example those of the Portuguese community in England, and some old Sephardic Siddurim like Tefilat HaHodesh or Bene Tsion, still retain the version that includes “shel”.
According to the Ashkenazi custom, the auxiliary candle or shamash should be lit first, after you say the Berakha, and with the shamash-candle you light the other candles. The Sephardic tradition is to light all the candles first, with a match or an additional candle, and the shamash is lit at the end, once all other candles are lit. In the Ashkenazi tradition, the shamash is used to light the candles. In the Sephardic tradition, no. In both cases the main reason why a shamash-candle is lit, is to avoid having a benefit from light of the other candles, if involuntarily we use the lights of the Hanukkia. (Perhaps this difference is also due to the fact that Sephardim usually use oil and Ashkenazim candles, and it is virtually impossible to light other candles with an oil candle…. )
Play with the dreidel (a four-sided spinning top) is an Ashkenazi custom that Sephardic children never practiced. Same, with Hanukka Gelt (coins or gifts for the kids).
In Sephardic communities it is customary to light one Hanukkia (or menora) per family. In many Ashkenazi communities, the custom is that each family member lights his or her own Hanukkia, even a boy or a girl once they are older than six (Rab. Eli’ezer Melamed).
Something similar happens regarding Shabbat candles: while according to the Sephardic tradition only the mother lights Shabbat candles, according to the Ashkenazi tradition, the daughters also light their own Shabbat candles.
Following the illustration mentioned two days ago: according to the Sephardic tradition, if a son or daughter, for example, are studying in Israel, it would not be required for them to light their own Hanukkia when they depending on their parents, because they would be included in the Mitsva of hadalaqat nerot Hanukka performed by their parents. According to the Ashkenazi tradition, however, a student living in his own apartment, must light his or her own Hanukkia with Berakha (even if she still depends on her parents).
This is a great video of the Story of Hanukka, produced by an educational department of the IDF. It is in Hebrew with Spanish subtitles (if someone finds it with English subtitles, please, send it to me!).