Why did Eastern Orthodoxy develop an emphasis on Christian mysticism while Roman Catholcism developed an emphasis on scholasticism? Did this predate the great schism?

One reason that Scholasticism didn’t catch on in the East (though it started there) is the trials in the 1070s-1080s against Platonists accused of corrupting Christian thought (John Italus and Michael Psellus), plus the general Byzantine anti-Roman bias and focus on mysticism and charismatic figures, etc. Ever since the Council of Sardica (342), and probably even earlier, it's clear that there was a different style/approach to theology in the East and West (in terms of understanding papal primacy, various ways of expressing theological truths, etc.); in the first thousand years this was largely seen as complementary by both sides, but of course by the Schism there was a more polemical attitude. Post-Schism scholarly humanists in Byzantium were not readily accepted, at all; the negative attitude toward Greek philosophy was mainly because Greek philosophy in the East tended to be Platonism ... Aristotle is more easily folded into Christianity, whereas Platonism has tended to be used to support heresies (Gnosticism, etc).

Byzantium only accepted the fourth-seventh century synthesis of the fathers and pagan thought, but never the new synthesis found in Catholic scholasticism. It's noteworthy, however, that Orthodox seminaries used neo-Scholastic manuals in the 1500s, 1600s, 1700s (until the modern Orthodox ressourcement movement, the Philokalia, etc.), and Catholicism, like most religions, has had mystics for its entire history ... they just didn't teach in the schools at Paris and London. At the time of Aquinas, Scotus, Ockham, etc., you still had plenty of Catholic mystics who were popular among the laity.

In any event, the humanists in Byzantium (who were open to scholasticism) were fully Christian, but every year all theologians were forced to repeat an anathema against Hellenic sciences. The typically uneducated Athonite monks who actually controlled Orthodox bishoprics and culture rejected Greek philosophy as “the deceiving appearance of true wisdom” (Meyendorff). The famous debate between Palamas and Barlaam led to a stronger emphasis on hesychasm/mysticism (versus scholasticism) and unfortunately “the victory of the monastic party in 1347-1351 simply implied the reaffirmation of a watchdog policy of the church against possible resurgence of secular humanism” (Meyendorff again), even if such secular learning was used for good purposes. Orthodox tend to wrongly see scholasticism as looking to nature in Aristotle’s sense (self-contained, divorced from nature as created by God) rather than natural man as the man in Christ. See Lubac on this issue, however; it's an Orthodox misunderstanding of the premises of scholasticism.

It was only in the late fourteenth century that we begin to see the translation of great Western scholastic works into Greek, used by many Byzantine theologians against Islam. Cydones was particularly important here (Meyendorff):

The religious evolution of Demetrius Cydones was determined by his discovery of Thomism as a philosophical system. As he himself clearly states in his writings, he suddenly discovered that the Latin West was not a barbarian land of “darkness,” as Byzantine humanists since the time of Photius thought it to be, but a new dynamic civilization, where ancient Greek philosophy was prized more than in Byzantium itself. He then began to castigate his compatriots for considering that in the whole universe there are only Greeks and barbarians ... that Latins in particular are never able to rise intellectually above the requirements of the military or merchant professions, while in fact so many Latin scholars were dedicated to the study of Plato and Aristotle ... “They show great thirst for walking in the labyrinths of Aristotle and Plato, for which our people never showed interest.” ... Thomas Aquinas, and certainly also the Italian Renaissance, were more “Greek” than Byzantium, especially since the latter has been taken over by the heyschasts [i.e. mystics]. ... Who can blame him for discovering that the classical Hellenic heritage was better kept in the universities of the Latin West than among the religious zealots of Mount Athos?

It's also worth noting that Greek theology was very "cultural" in the sense that all educated people knew it, and there were a lot of educated people in Byzantium; civil servants, random people on the street, etc., which led to cultural conservatism, more resistance to scholasticism and development. In the West, the schools were separate in a cultural sense (using Latin versus vernacular, etc.), and hence had more independence. Byzantine monks were usually in the cities, and virtually all bishops were originally mystics/monks, which led to a very strong emphasis on mysticism and the traditional pre-scholastic synthesis of philosophy/theology found in the first thousand years. See for example (quoted in Nichols) a Byzantine theologian in 1429:

I too, if I wanted, could have sophistical arguments with syllogisms better than yours. But I do not want. I ask my proof from the fathers and their writings. You will oppose to me Aristotle and Plato, or perhaps your recent doctors; against them I put the sinners of Galilee with their frank speech, their wisdom and seeming madness.

In a sense you could say that Orthodoxy becomes Pietism, a curiously modern rejection of discursive rationality as used in scholasticism. This leads to Orthodox readings of Catholic scholasticism and mysticism that are ... problematic (Patriarch Bartholomew):

Western scholasticism reduced divine truths and god himself to mere concepts. Communion with god is a conceptual enterprise. And so the genuine experience of god, beyond concepts and ideas, the participation in the uncreated light and, generally, the uncreated energies – all of this becomes an impossibility for the west.

This is, in my opinion, completely wrong; Catholicism still has mysticism and there are plenty of ways to describe the experience of God without the theologically vague and problematic concept of essences/energies of God (also, as A. N. Williams shows, Aquinas and Palamas were saying roughly the same thing, just in slightly different terminology). As one Orthodox commentator puts it (Henry Reardon):

What almost always passes for Orthodox theology among English-speaking Orthodox these days is actually just a branch of the larger Orthodox picture. Indeed, it tends sometimes to be rather sectarian. The Orthodox Church is an ancient castle, as it were, of which only two or three rooms have been much in use since about 1920. These two or three rooms were furnished by the Russian émigrés in Paris between the two World Wars. This furniture is heavily neo-Palamite and anti-Scholastic. It relies heavily on the Cappadocians, Maximus, and Gregory Palamas ... Anything that does not fit comfortably into that model is dismissed as Western and even non-Orthodox ... this popular neo-Palamite brand of Orthodoxy, though it quotes Damascene when it is convenient, never really engages Damascene’s manifestly Scholastic approach to theology. Much less does it have any use for the other early Scholastic theologians, such as Theodore the Studite and Euthymus Zygabenus. There is no recognition that Scholasticism was born in the East, not the West, and that only the rise of the Turk kept it from flourishing in the East.

I could say quite a bit more but the character limit is rapidly approaching ... I guess this covers most of the basics. Source: I have a doctorate in theology and have spent years studying this precise topic (I'm Orthodox and my wife is Catholic). The best book on this topic is almost certainly Aidan Nichols, "Rome and the Eastern Churches," but there's also good historical work from Yves Congar, John Meyendorff, etc.

edit1: formatting

edit2: maybe an even simpler rubric would just be to look at the general tendency in Christianity to have varying degrees of accommodation of pagan culture/philosophy ... to give modern examples, at one extreme you have the early Luther (rejecting virtually all pagan philosophy), and on the other end, Hegel (who literally subsumes/sublates Christianity within philosophy, stating in his letters that he considered himself a Christian theologian and in his lectures on the philosophy of religion that he saw himself as the heir to the Scholastics, completing their project). The Orthodox rejection of, and Catholic embrace of, Scholasticism are closer to the middle of this spectrum ... but there have been varying emphases on "mysticism-charismatic-subjectivity" versus "rationalism-theology-scholasticism," kataphatic versus apophatic theology, etc., throughout the history of Christianity.