The Melkites are Byzantine Catholics from the Near East (Syria and Lebanon, mostly).
Melkites have their own Patriarch: http://www.melkite.org/PatriarchWritings.html
The Byzantine Catholics, as they are commonly referred to in the US, are Ukrainians, Ruthenians and Slovaks from Eastern Europe, while the Melkites are exclusively (originally of course) from the Middle East. They share essentially the same liturgy, but then again, so do most Eastern Christians. There are several churches which are lumped together as “Byzantine” because of their Greek basis and liturgical background: Melkite, Romanian, Ruthenian, Russian, Slovak, Ukrainian, and Italo-Greek (Old Albanian?).
Again, however, the Melkites have their own jurisdiction, and their own patriarch.
When you see a “Byzantine Catholic” parish it is not going to be a Melkite parish. That would be listed as a Melkite parish: http://www.melkite.org/
Example: Melkite: http://www.melkite.org/parishinfo.html
Byzantine: http://www.stannbyz.org/
and the same name, but different county in Pa. is a St. Anne’s that 25 years ago was called “Byzantine” but now is said to be “Ukrainian Catholic”: http://www.stanneukrainiancc.com/