I'm thinking, I suppose, of the period say 500-1400 or so, which I realize predates the "official" split between the churches. In particular, I tend to think of eastern/Orthodox monasteries as being much more interested in hesychasm and meditative practices than western monasteries, and more detached from the world, but I don't know if that's true or not.
One of the defining characteristics of Eastern Monasticism is its interest in solitude. The first monks (fourth century) in the East were the Desert Fathers who went into the Egyptian desert to become hermits. Their teaching have been studied and preserved in "The Sayings of the Desert Fathers". While eastern monasticism went on to found many monasteries, this hermit tradition has been maintained. Even in the 20th century, many monks chose to live the last years of their life in isolation.
Mount Athos (a peninsula in Greece) became the center of Orthodox Monasticism. A large concentration of monasteries was founded there, not just from Greece, but eventually other Orthodox countries such as Russia and Bulgaria. Interestingly, the monks on Mount Athos did not suffer greatly under Ottoman rule. They shrunk considerably during the Post-War period when many traditionally Orthodox countries were ruled by Communists, and much of the support they received was reduced.
Catholic Monastics during the medieval era were not dramatically different. Both received large donations from rulers and nobles. One main difference is that the Catholics founded a series of religious orders over time. Orders such as Benedictines had a large number of monasteries which followed the same set of strict rules.