There are no major differences between the Sephardic and the Ashkenazi traditions for the celebration of Hanukka, just a few minor variations.
Some of them are:
1. The Ashkenazi tradition is to say the Berakha: "lehadliq ner shel Hanukka" while most Sephardim would say: "lehadliq ner Hanukka", without the word "shel." It is interesting to know that, although there is not semantic difference between the two versions, the original version of the berakha (as per Maimonides MT Hanukka 3:4 ) is "lehadliq ner shel Hanukka". Sephardim from Spanish Portuguese communities still preserve this version.
2. In the Ashkenazi Minhag, the auxiliary candle (shamash) is lit first and with it one lights the rest of the candles. The Sephardic Minhag is to light all the candles first, with a regular match or candle, and the shamash is lit at the end. In this last case, the shamash acts as an auxiliary candle not for being used to light with it the other candles but for avoiding benefiting from the light of the Hanukka candles.
3. Playing with the dreidel, spinner or sebibon is an Ashkenazi custom that Sephardim never practiced. Same as Hanukka Gelt (money or gifts to the children).
4. In Sephardic communities it is customary to light only one Hanukkia per family. In Ashkenazi communities the custom is to light one Hanukkia for each member of the family. Following the Ashkenazi tradition, for example, a student who lives in his own apartment would light his or her own Hanukkia with Berakha (even if he is still depending on his parents). Incidentally, this is also the case regarding Shabbat candles: while according to the Sephardic tradition only the mother lights Shabbat candles, following the Ashkenazi tradition the daughters would light their own candles, saying Berakha for it.