The differences are indeed fascinating.
As usual I agree with markos.
There is a much a person could say on this topic.
I'll only say a few things.
One of the biggest differences is the role and style of hymnody.
Hymnody in the west and Hymnody in the east took significantly different forms.
Much of the hymnody of Gregorian chant is less known today. What is typically heard from it is based primarily on psalms.
For Byzantine chant it is just the opposite, most of what is heard today is more so hymnody and less psalms or scriptural and in fact within the divine office many canticles formerly sung from the old and new testament have been replaced with odes and canons around the 8th century.
I think it is fair to say that the western liturgy in general is a survivor of a more primitive style, but also of course different cultural mindset for music. However part of this feeling is also because it has had much of it's hymnody in the forms of tropes and sequences was officially removed from it's important feast days during the reformation, thereby giving it a more sparse quality.
In the west hymnody was largely metrical. a certain amount of non-metrical hymnody exists in the ancient west (I can think of the hymn for the feast of st ursula and the 10,000 virgins for example) but I have not encountered non metrical hymnody very often.
In the east the use of metrical hymnody in the liturgy of the church was taboo from and early date and associated with pre-christian greco-roman-semitic pagan poetry. So it never developed with strict metre. 'Aghia Parthena', composed in the early 20th century by St Nectarius is an example of a metrical hymn from byzantine culture which is ment to be used outside of liturgy.
In the west hymnody in came in about 4 forms. The #1 form was of divine office hymns, #2 form was sequences/proses intended exclusively for the mass, which came into full fruition by the 800s.
(l) Sequences (sequentioe, prosoe, prosa). These are the artistically constructed songs, consisting of strophe and counterstrophe, inserted in the Mass between the Epistle and the Gospel.
The sequences were based in their origin in the 700s on the melody of the melisma of the alleluia chants, though by the 800s they were no longer always based on the alleluia melodies. Originally sequences were had only one note per syllable and used a more free type of metre (making them more challenging to sing), but gradually by the 12th century the sequences being composed for new feasts and saints became more regularly metrical and at times simpler, though they more often used more than one note per syllable and were more greatly exploiting the full human vocal range by that time. (1 and 1/2 octaves?).
The #3 form was conductus and processional hymn (or processional antiphon) (various names I cant remember), which tended to repeat the same words over and over with different verses inbetween. Some of the "final antiphons of the blessed virgin mary" were originally sung this way. sometimes sequences were used at processions during the office.
A good example of this type of hymn to survive to the present day is the Gloria Laus et Honor for Palm Sunday.
The Salve Regina shown here is another example: http://www.youtube.com/watch?v=IoSuyUFiEYo
No one EVER sings the Salve regina with verses inbetween anymore today.
for the 4th type I will simply paste the quote from the catholic encylcopedia:
Hymnody of the missal or the gradual
(2) Tropes of the Mass (tropi graduales). During the Middle Ages, all those parts of the Mass which were not sung by the priest but by the choir, e.g. the Kyrie, Gloria, Sanctus, Agnus Dei (tropi ad ordinarium missoe) also the Introit, Gradual, Offertory, Communion (tropi ad proprium missarum) were provided with a rich setting of interpolatio, more even than the Breviary. These tropes came to be known as "Tropus ad Kyrie", "Tropus ad Gloria", etc. or "Troped Kyrie", "Troped Gloria", and so on.
One of the reasons why it is easier to adapt hymns of byzantine chant to various languages is because it is not based strictly on meter. This may prove to be a weakness for latin hymnody. Though a large portion of it has been put into english by anglicans, there is an inherent weakness in that the translations are not as literal as they could be due to the extent that paraphrase must be used at times to achieve metre in english.
The last comment I have is that I see strong similarities in the melody of the Glory in the Highest to God (Gloria in Excelsis Deo) "Great Doxology". The Latin melody for Mass setting II (fons bonitatis) in the Graduale Romanum is striklingly similar to that heard in several greek versions. Both of these were sung originally in Matins/Orthros east and west, so this is not surprising. Most of the other melodies are more different, but I continue to see resemblences across the board for the Great Doxology in both latin and greek melodies.