[There is evidently a great deal of interest in the various rites of the Church, and in particular the lesser known Western rites. As such, I've determined to post some quick summaries of some of the basic differences, by simply pulling them off extant sources like the Catholic Encyclopedia. It is difficult to do much more without the original liturgical materials in front of you after all.
It's worth noting that in my piece recently on the Divine Office, I proposed an over-inflated idea of Romanitas has homogenized our Western liturgical rites and uses, both in the Missal and in the Breviary. (This should not be taken as a polemic against the See of Rome, which we indeed venerate.) Take note in this summary -- which I've re-arranged and taken from the Catholic Encyclopedia via Wikipedia -- the actions of General of the Cistercian Order, Claude Vaussin, in the middle of the 17th century.]
A Summary of the Cistercian Rite and its variances from the Roman
Prior to the 17th Century:
The Missal:
Prior to its later reform (in the middle of the 17th cenutury) there were wide divergences between the Cistercian and Roman rites.
i) The psalm "Judica" was not said, but in its stead was recited the "Veni Creator"; the "Indulgentiam" was followed by the "Pater" and "Ave", and the "Oramus te Domine" was omitted in kissing the altar.
ii) After the "Pax Domini sit semper vobiscum", the "Agnus Dei" was said thrice, and was followed immediately by "Hæc sacrosancta commixtio corporis", said by the priest while placing the small fragment of the Sacred Host in the chalice; then the "Domine Jesu Christe, Fili Dei Vivi" was said, but the "Corpus Tuum" and "Quod ore sumpsimus" were omitted.
iii) The priest said the "Placeat" as now, and then "Meritis et precibus istorum et onmium sanctorum. Suorum misereatur nostri Omnipotens Dominus. Amen", while kissing the altar; with the sign of the Cross the Mass was ended.
Outside of some minor exceptions in the wording and conclusions of various prayers, the other parts of the Mass were the same as in the Roman Rite.
In some Masses of the year the ordo was different; for instance, on Palm Sunday the Passion was only said at the high Mass, at the other Masses a special gospel only being said.
The Missal from the 17th century and onwards:
Under Claude Vaussin, General of the Cistercians (in the middle of the seventeenth century), several reforms were made in the liturgical books of the order. Since then, the differences between the Roman and Cistercian rites are not as substantial.
The Breviary:
Quite different from the Roman, as it follows exactly the prescriptions of the Rule of St. Benedict, with a very few minor additions. St. Benedict wished the entire Psalter recited each week; twelve psalms are to be said at Matins when there are but two Nocturns; when there is a third Nocturn, it is to be composed of three divisions of a canticle, there being in this latter case always twelve lessons. Three psalms or divisions of psalms are appointed for Prime, the Little Hours and Compline (in this latter hour the "Nunc dimittis" is never said), and always four psalms for Vespers. Many minor divisions and directions are given in St. Benedict's Rule.