The difference between the Hanukkiah and the (Temple) Menorah lies in the number of candles each holds as well as the purpose and time of year each is used. The Hanukkiah is a type of menorah or candelabra that holds 9 candles and has 8 branches. The Menorah is a similar candelabra with branches.
In English, menorah was originally the name for the seven-branched candelabra used in Jewish worship. The nine-branched Hanukkah candelabra is called hanukkiah in Hebrew, but English speakers came to use menorah for this too. The Hanukkah menorah recalls expulsion by Judah Maccabee of invading forces from the Temple of Jerusalem. Maccabee and his followers sought oil for the temple's menorah so that the sanctuary could be rededicated, but they found only enough oil for a single day. Miraculously, that tiny amount of oil burned for eight days, until a new supply could be obtained. The Hanukkah menorah includes a candle for each day the oil burned, plus the shammes, a "servant candle" that is used to light the others.