The antiphons (it is good to give thanks.... through the prayers) are selected different psalm verses. These were supposedly part of the "Cathedral" liturgy used in Hagia Sophia and the Ecumenical Patriarchate used them to replace Psalm 102, 145 and the Beatitudes after that liturgy was "rediscovered" in the 1800s. The replacement was supposed to be more participatory for the people.
I don't think this could be the case because if it were these "rediscovered" antiphons would not have made thier way back into the Byzantine Catholic DL as these Churches were already reunited with Rome by 1800.
I have often wondered about the difference in antiphons myself. The Slavic Churches seem to use the "Bless the Lord, Oh my soul," "Praise the Lord, Oh my Soul," and the Beattitudes, while the Middle Eastern/Greek Churches seem to use the "Through the Prayers of the Theotokos, Oh Savior Save Us." The exception is the Ruthenians, who although Slavic, currently use the "Through the Prayers of the Theotokos, Oh Savior Save us," but I believe this was only instituted post Vatican II and prior to that they used the traditional Slavic antiphons.
Below is an excertps from THE EVOLUTION OF THE BYZANTINE LITURGY by Juan Mateos, S. J.
"THE THREE ANTIPHONS
Three antiphonal psalms were sung in Constantinople not only at solemn processions, but also at the Vesperal Office and other liturgical services. During the processions, the singing of the antiphons took place at a stop or station. These three antiphons were independent of the processional Troparion and psalm sung while the procession marched to the church.
We saw already that the Trisagion with its psalm, originally a procession chant, became the first Introit or entrance hymn of the Liturgy, and that in more recent times, variable entrance hymns were composed, without suppressing the Trisagion.
Between the seventh and eighth centuries a new development took place in the Liturgy: the preparation of bread and wine, which until that time had been done after the lessons and litanies (that is, at the beginning of the Eucharistic part of the Liturgy), was transferred to the very beginning, before the first entrance. While this rite of preparation was going on, it was necessary to fill the time with some pious exercise, and three antiphons, similar to those sung in the Forum on certain days, were added to the beginning of the Liturgy. The already existing entrance hymn became the third antiphon, thus losing its function as an Introit, and the two new antiphons were placed before it.
Thus for the third antiphon or entrance hymn, the old processional Psalm 94 (95), «O come, let us sing to the Lord … let us come into His presence with thanksgiving,» was kept. For the first and second antiphons two preceding psalms were chosen, Psalm 91 (92), «It is good to give thanks to the Lord»; and Psalm 92 (93), «The Lord reigns; He is robed in majesty.» Psalm 93 (94), «O Lord, Thou God of vengeance,» hardly suitable for a joyful celebration, was happily avoided!"
This the full article is really supurb and is available at:http://jbburnett.com/resources/mateos_evolution_of_byz_ltg.pdf