There are no major differences between the two traditions, just a few minor variations as follows:
1. The Ashkenazim say in the Berakha: Lehadlik ner shel Chanuka. In the Sepharadic Minhag: lehadlik ner Chanuka, without the word "shel." The Mashadi tradition, however, is to say 'shel' following Bene Tziyon Siddur.
2. In the Ashkenazi Minhag, one first lights the Shamash (the auxiliary candle) and with the Shamash one lights the rest of the candles.
In the Sepharadic Minhag all the candles are lit first, with a regular match or with a longer candle and the Shamash is lit at the end. Also in this case, most Mashadi families I know would follow the Ashkenazi practice.
3. In the Sepharadic Chanukia the Shamash is placed, normally at the end of the Chanukia, In the Ashkenazi Minhag, in the beginning or in the middle, and a little higher than the rest.
4. For Sepharadim, it is customary to light only one Chanukia for all the members of the family. In many Ashkenazi communities they light one Chanukia for each member of the family.
Incidentally , this is also the case regarding Shabbat candles: while according to the Sepharadic Minhag only the mother lights the candle, in the Ashkenazi Minhag the daughters also light their own candle and say Berakha for it.
5. The famous Dreidel, spinner or sebibon is originally an Ashkenazi custom, which Sepharadim did not use to practice in the past. Same as Chanuka Gelt (money or gifts to the children).
More on Sepharadic and Ashkenazi traditions:
This article is a Chanuka gift for all those who love Israel!!!
Rabbi Yosef Bitton. YMJC 130 Steamboat Rd. Great Neck NY 11024