Byzantine Chant vs. Gregorian Chant: What are the differences in goals as far as attitude and concepts between these two chanting styles 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.
This is purely subjective, and others may feel differently. I'm not claiming 
this to be an absolute truth, but here it goes:

I believe that Catholic theology is centered around exaltation and Orthodox theology is centered around humbleness.

I see this in the different architectural approaches. Catholic churches with their tall, narrow and pointy windows instill a sense of exaltation, of going upwards towards the Heavens while the huge central Pantokrator icon of Christ on the ceiling of every Orthodox church is extremely humbling.

The iconographies speak the same thing to me as well. The Catholic canonical depiction of the Resurrection depicts Christ triumphantly rising from the grave, sometimes alone sometimes with two angels bowing down to Him, again, very exalting. The Orthodox canonical depiction of the Resurrection shows Christ rising triumphantly from a defeated Hell (gates and keys broken), dragging Adam and Eve out of their graves, again, humbling, as we, the sinners, identify with Adam and Eve.

And finally, I perceive Gregorian chanting to be exalting and Byzantine chanting to be humbling.

However, I may be wrong, the perception of music is a huge academic field, but I've pointed out the unrelated examples of architecture and iconography to validate my (again, subjective) consistency.