This has been well covered on the Internet, including message boards, in the decade I've been online but in short
After the mediÃƒÆ’Ã†â€™Ãƒâ€šÃ‚Â¦val estrangement of the Christian West and East there were reunions with Rome of parts of the Orthodox (and the big reunion at the council of Ferrara-Florence that fell through after the Turks overthrew the Byzantine Empire). Those parts are the Byzantine Catholic churches.
These partial unions happened for various reasons, such as Greeks and Albanians living in Italy, avoiding Polish persecution (the Ukrainian Catholic Church started in 1596) or Hungarian Protestant persecution (the Ruthenian Catholic Church started in 1646), Roman Catholic work for individual conversions* (the Melkite Church in 1724, the very tiny Greek Byzantine Catholic Church circa 1900), people on their own reading their way into it (the Russian Catholics) and in one case, nationalists looking for a better deal - their own national church - than they got at the time from the Orthodox (the Bulgarian Catholic Church in the late 1800s).
How do they differ from the Orthodox today?
They're under Rome.
And most of them have disobeyed Rome and latinised themselves - trading their own rite's customs for Roman Catholic ones. (Exceptions: the Melkite Church, which is the analogue to the Antiochian Orthodox, and the tiny Russian Catholic Church, the 100-year-old analogue to the Russian Orthodox.) They're supposed to be just like the Orthodox but often aren't.
In North America they're not allowed to ordain married men. (Rome's concession since 1929 to American RC prejudice against the BCs - it made some BCs switch to the Orthodox.)
They're smaller churches than the Orthodox.
Hope that answers your question.
*Which they don't do anymore - the goal now is corporate reunion, not individual conversion at the expense of the Orthodox using the BC churches.