I'm NOT interested here in the difference in musical structure and rules here, per-say, between the Byzantine and Gregorian chanting styles.
What I'm more interested in is the difference between them in the goals of expression in each chanting tradition?
For example, what is Gregorian mode 1 trying to convey compared to Byzantine mode 1?
Or also, more generally, what is Gregorian Chant as a whole trying to accomplish compared to what Byzantine Chant?
When I say Byzantine Chant, I mean the Greek Byzantine Chanting tradition used by the Greek Orthodox Church on Mt. Athos and throughout the Churches in communion with Constantinople.
When I say Gregorian Chant, I mean the Gregorian Chanting tradition used by the Roman Catholic Church that is most often associated with Gregorian monastics.