Hesychasm, one of the monastic traditions this question is about, is still practiced on the Greek peninsula, Mount Athos. For over a thousand years, monks have practiced the discipline of continual prayer and worship, which goes back at least as far as the 4th century Desert Fathers. And of course, the concept of praying continually goes back at least to Paul:
Rejoice always, pray continually, give thanks in all circumstances; for this is God’s will for you in Christ Jesus.—1 Thessalonians 5:16-18 (NIV)
The idea of going to a private place to pray (becoming a hermit in extreme practice), goes back to Jesus himself:
But when you pray, go into your room, close the door and pray to your Father, who is unseen. Then your Father, who sees what is done in secret, will reward you.—Matthew 6:6 (NIV)
From the outside, there does not seem to be too many differences between Western and Eastern Monasticism. But since the Great Schism (and obviously even before), the two great branches of Christendom have diverged in doctrine and praxis. To take a small instance, opponents of Hesychists accused them of practicing Omphaloskepsis (literally "navel gazing") because of one of their prayer postures.
More seriously, Eastern traditions place a greater emphasis on icons than Western traditions. (Some Protestants go as far as to abolish all images in their places of worship.) Due in part to Saint Basil the Great, the Eastern Orthodox traditions place much greater importance on images to help focus worship and prayer. Perhaps the greatest treasure of Mt. Athos is their extensive collection of Byzantine painting.
On the other hand, Mt. Athos was not the center of scholarship that many Western monastic centers were (and are). Gregory Palamas famously defended Hesychasm against the scholastic Barlaam of Seminara. By this point, Eastern Monastics had largely rejected secular teaching as unprofitable for drawing closer to God. In later years, the Eastern monastic tradition has grown more separate from the secular world, while Western monastics have maintained contact. As an example, females (both human and animal!) have been forbidden to travel to Mt. Athos.
Even so, Hesychasm continues to have a large influence in Eastern Orthodoxy. Devout believers still travel to Mt. Athos to observe and be a part of the continual prayers and worship going on day and night in the various monasteries. It is held that these men have a holy connection to God and stand on the spiritual front-lines against evil. Novices take on vows to remain on the peninsula for the rest of their lives in order to draw closer to God.
Sources for this answer include Sailing from Byzantium: How a Lost Empire Shaped the World by Colin Wells and an episode of 60 Minutes. Wikipedia has good articles on specific people and belief systems, which greatly helped me recall details. Not directly part of the answer, but fascinating to me, is an article in Vanity Fair by Michael Lewis on the monks' role in the Greek financial crisis.