Aside from the mention of the Pope during the commemoration of hierarchs in the Anaphora, the Divine Liturgy of Greek rite Catholics and Byzantine Orthodox really only has one major distinction in the United States: the Catholics say the Anaphora aloud more often than not, aside from some more ethnic parishes which tend to emulate the Motherland where the Orthodox praxis persists happily. However, there is one Byzantine Catholic rite that resembles the Orthodox and Greek Catholic liturgies just a little less; that would be the Ruthenian Catholic Church's Divine Liturgy.
The Ruthenian Church, initially comprised of Catholic immigrants from the Carpatho-Ruthenian mountains to America, follows the Slavic branch of the Divine Liturgy (except for the Little and Great Entrances, which circle the entire temple). Their Liturgy possesses the same order of Litanies found in other versions of the Divine Liturgy, but the concluding sacerdotal blessing, which is normally said silently by the priest if a deacon leads the litanies or omitted by the priest if there is no deacon, are always said and always said aloud.
Why? Why must I hear the priest, after the Litany of Fervent supplication, pray aloud
"Lord our God, accept this fervent supplication from Your servants, and have mercy on us in accordance with the abundance of Your mercy, and send down Your compassion upon us and upon all Your people who await Your great and rich mercy"when I have just heard those very words sung in the Litany itself:
"O Lord almighty, God of our Fathers, we pray you, hear and have mercy.
"Have mercy on us, O God, according to your great mercy, we pray you, hear and have mercy."Aside from the obvious redundancy this repetition ignores an important element in the Divine Liturgy's distinction between the role of the priest and the deacon, which is that the priest serves Christ's priesthood before God while the deacon is the leader of popular prayer. The received praxis of the priest and deacon saying this prayer in differing forms at the same time unites God and the people through their respective offices. Repeating this prayer aloud foregoes this essential point in favor of wordiness.
As mentioned before, the Ruthenians follow the general tendency of Greek rite Catholics in the United States to sing the anaphora aloud. The Ukrainian and Melkite Churches follow this rule for the Liturgy of Saint John Chrysostom, but for the Liturgy of Saint Basil, with its very, very long anaphora and commemorations, they will continue to recite it while the choir sings the responses (ex. "We praise You, we bless You, we give thanks to You, and we pray to You, O our God!"). Not so for the Ruthenians, who will sing every word aloud as was done in 10th century Constantinople on every Sunday after Pentecost. This ought not be criticized on its own, but it certainly removes the veil of revelation and mystery that accompanies the consecration in the Divine Liturgy, with the calling down of the Holy Spirit behind a curtain and the finalization of the change of the gifts in a manner unseen and unheard until the Sacred Species are brought forth for Holy Communion with the charge "Approach with the fear of God, with faith and with love."
The strangest element of the Ruthenian Divine Liturgy, at least as practiced in the two Ruthenian parishes I have attended in Texas, is that the clergy communicate simultaneously and in a manner that assimilates their Communion with the people's. Normally the celebrant takes the Holy Bread, then any concelebrants, then the deacons receive from the priest and consume the Sacred particle on the south side of the altar; a similar order follows for Communion by the chalice. The Ruthenian rite follows this pattern for Communion of the Holy Bread, but the clergy do not actually eat the Body of Christ until after the congregation has finished the prayer "I believe and confess that You are truly Christ, the Son of the Living God...." Then follows clerical Communion from the Chalice and the people's Communion, albeit preceded by the aforementioned "Approach with the fear of God".
There are other curious elements, like the abbreviation of the antiphons just to include one line of "Through the prayers of the Theotokos, O Savior save us" and "O Son of God, risen from the dead, save us who sing to You, Alleluia" rather than the fuller psalmody. And then there is final blessing, so often rendered as "for He is good and He loves mankind" or "for He is good and the Lover of Mankind." The Ruthenian Church here offers "for Christ is good and He loves us all."
This revision of the Ruthenian liturgy was fairly recent (2006). One hopes that they might re-consider aligning with the norms of the other Greek rite Catholic and Orthodox Churches.