I've heard the term Greek Catholic. Historically the Byzantine Catholic Church used to be known as the Greek Catholic Church. In Europe, the term Greek Catholic is still the norm. Any Catholic can attend any other Catholic Church to fulfill their obligation. Icons are used instead of statues.
Incense is used throughout the Liturgy.
Byzantine Churches have large Icon Screens which separate the Altar area from the rest of the church. Byzantine priests can be married in Europe, but currently not in the U.S.A. or Canada. We use the Liturgy of St. John Chrysostom, which dates back to the 300's. The Liturgy has also always been in the common language of the people. I'd like to become a Byzantine Catholic.