Hi Matt,
Eastern Catholics belong to a variety of Catholic Churches, each headed by a Patriarch, as the Latin Church is headed by a Patriarch (the Pope) who, in this case, happens to be both a Patriarch and the Head of All Catholics, including Eastern Catholics.
Byzantine Catholics are one of a group of Eastern Catholics.
All Catholics believe in the same truths. Where the Eastern Churches differ is mainly in two fields : the liturgy and canon law. They have their own ways and languages for celebrating mass and other liturgical functions. They also have a separate code of canon law.
One must not confuse Eastern Catholics with Eastern Orthodox Christians who do not accept the authority of the Holy Father. Most Eastern Catholic Churches were at one time part of the Eastern Orthodox community, and they have come back to the fold.
If you do a web search with “Eastern Catholics”, you will find many interesting sites.
Verbum