How can you compare and contrast Christmas and Hanukkah?

Christmas and Hanukkah take place around the same time of year. That is the only real similarity.

Most other similarities are the result of Christmas influencing Hanukkah, such as gift giving and decorations.

Christmas is a celebration of the birth of Jesus, hence the nativity scene featuring baby Jesus that has been popularized. Christmas is a Christian holiday. People who believe in the New Testament and that Jesus died for their sins and was resurrected practice this holiday. Modern Christmas celebrations include getting and decorating a Christmas tree, singing Christmas or Christian carols and hymns, hanging stockings by the fire, telling stories about Santa Claus and his reindeer, and wearing red and green.

Christmas also contains numerous Pagan customs, as it is a hybrid of Pagan and Christian traditions.

Hanukkah is a Jewish holiday celebrating a post-biblical story in which the Greco-Syrian king forbade the Jewish people from practicing their monotheistic religion. The Maccabees (meaning 'hammer') were a small group of rebels, led by Judah 'Maccabee' and his sons, that defeated the Greco-Syrian army that had overtaken the the Holy Temple in Jerusalem. After the Jewish people won back their Temple, they held a delayed celebration of the 8-day harvest festival of Sukkot, which is the reason Hanukkah is 8 days long.

A story in the Talmud (500 years later) says that the Jewish people won back their temple and wanted to light their menorah to praise their God. But because the Greco-Syrians had trashed their temple, there was only enough oil for one day--the miracle is that it lasted for eight days. So Jewish people light an eight- (plus one [the 'Shamash' candle] which lights the others) branched Hanukkiyah (a candelabra) to celebrate this miracle. This is considered to be a legend with the purpose of adding a spiritual level to the holiday.