What is the difference between a Maronite church and Melkite church?

Both "churches" are currently Rites within the Catholic Church. The Maronite Rite is consists of hte nation and church of most Arabic-speaking Syrians living in Lebanon. There are also individual churches of this rite found in Palestine, Cyprus, Egypt, and the United States. They have been in communion with Rome continuously since the Fifth Lateran council, they use the Rite of St. James in ancient Aramaic in their liturgy.

The Melkites are Byzantine Christians who remained faithful after the Council of Chalcedon (A.D. 451) when large numbers in the near East accepted the Monophysite heresy. They were briefly involved in the Greek Schism from the ninth to the eleventh centuries, but have been reunited with Rome since the eighteenth century. They follow the Byzantine Rite, mainly in the Arabic language.

extracted from Modern Catholic Dictionary byJohn A. Hardon, S.J. Doubleday & Co., Inc. Garden City, NY 1980