How does the divine office compare between the different rites? I'm wondering how the liturgy of the hours looks different in different quarters of the church. When Byzantine Christians pray the Divine Office, how is it different to when Latin Christians pray the Divine Office? How about Copts? etc. What do they have in common and what is different between them?

If this question is too broad, please scope it to a comparison of only the Latin and Byzantine rites.
Commonalities between the Horologion (Eastern Divine Office) and the Divinum Officium (Roman Office):


Psalms are all found, that is assured.


Some differences between the Divinum Officium and the Horologion:


The Hours in the Eastern Church often are made up of a variety of different books. Whereas the Hours of the Roman Church can be used within a single book called the Roman Breviary, plus the Matins, usually separate, which contains the rest of the entire Sacred Scripture, papal homilies, and the Gospel.
The Psalter in the Byzantine Office is divided into 20 kathismata, where each is divided into three stases each. Whereas in the Roman Office, the psalms are distributed across the span of (not 3, but) 4 weeks in the Liturgium Horarum, and in a span of 1 week for the 1962 Divine Office and the pre-1955 Divine Office.
The rubrics of the timing of the recitation of the Hours differ between the Horologion and the Divinum Officium.

In the Horologion, The Ninth Hour, Vespers, and Compline (Apodeipnon) may be said in block before sunset, calling it the Evening Aggregate. The Midnight Office (Mesonyktikon), Othros(Matins), and the First Hour can be recited in the morning before sunrise. The Third Hour and the Sixth Hour are recited in the midday, calling it the Midday Aggregate.

In the Roman Office however, Matins is typically said with Lauds in the morning, then Vespers with Compline in the evening, then the person typically recites the other Hours--Prime, Terce, Sext, and None separately throughout the day. There are not much governing rubrics as to mind when the Hours may be recited, except that:


In the pre-1955 Office, Matins and Lauds may be anticipated as early as 2 PM before.
In the pre-1969 Office, Matins may be anticipated without Lauds in the evening after Compline of the Day
In the Liturgium Horarum, you can recite the Office of the Readings any time of the day.

The Canticles (Benedictus, Nunc Dimittis, Magnificat) are said in different Hours:


The Magnificat is said at Vespers in the Roman Office, meanwhile it is said in Matins in the Eastern Offices
Benedictus is not sung as a canticle in the Eastern Office, but it is the Gospel reading for Matins and the Divine Liturgy on the Feast Day of the Nativity of John the Baptist (June 24). It is a Canticle however, at Lauds
Nunc Dimittis is sung every Compline in the Roman Office, whereas in the Eastern Offices it is sung at Vespers.



The differences in the Divine Office between the two churches are vast; depending on the version of the Offices in question, the liturgical season, day, religious orders, and even the number and status of the participants.