Why do Greek Orthodox venerate icons and Roman Catholic statues? Historically how and why did this difference arise?
Why do Greek Orthodox venerate icons and Roman Catholic statues?

This question is quite nicely phrased on a couple points of view. Before going on let me mention that the Eastern Orthodox Church generally venerate icons. But in the West, Roman Catholics tend to venerate statues. I would like to point out that Eastern Rite Catholics do as the Eastern Orthodox do and venerate icons. 


  The seventh Ecumenical Council of Nicaea (787) justified the veneration of icons of Christ, his mother and the angels and saints. The ultimate justification of images is God's incarnation in Christ: by taking a human body and nature, God made himself visible and material. The honor paid to sacred images is a "respectful veneration," not the adoration due to God alone. Christian veneration of images is not idolatrous, sacrilegious because the honor rendered to an image passes to its prototype, meaning to the person portrayed in the image. Images are mere things in themselves, but they are leading us on to God incarnate. The "bowing down" has a variety of meanings: in some cultures people like you and me bow before each other. Therefore it is important to know what the "bowing down" means. In the Catholic Church "bowing down" is contrasted to genuflection (kneeling). Generally, the latter indicates worship (God), the former is a lesser act on honoring somebody, i.e. Mary or a saint. - Veneration of Icons and Religious Images


Part of the reason why the East incorporated the veneration of  icons instead of statues is tradition and part is that the use of icons predated the veneration of statues in the West. Let us not forget that in the East, an icon is read.


  An icon (from Greek εἰκών eikōn "image") is a religious work of art, most commonly a painting, from Eastern Orthodox Christianity and Catholicism. More broadly the term is used in a wide number of contexts for an image, picture, or representation; it is a sign or likeness that stands for an object by signifying or representing it either concretely or by analogy, as in semiotics; by extension, icon is also used, particularly in modern culture, in the general sense of symbol — i.e. a name, face, picture, edifice or even a person readily recognized as having some well-known significance or embodying certain qualities: one thing, an image or depiction, that represents something else of greater significance through literal or figurative meaning, usually associated with religious, cultural, political, or economic standing.
  
  In Eastern Christianity and other icon-painting Christian traditions, the icon is generally a flat panel painting depicting a holy being or object such as Jesus, Mary, saints, angels, or the cross. Icons may also be cast in metal, carved in stone, embroidered on cloth, painted on wood, done in mosaic or fresco work, printed on paper or metal, etc. Creating free-standing, three-dimensional sculptures of holy figures was resisted by Christians for many centuries, out of the belief that daimones inhabited pagan sculptures, and also to make a clear distinction between Christian and pagan art. To this day, in obedience to the commandment not to make "graven images", Orthodox icons may never be more than three-quarter bas relief. Comparable images from Western Christianity are generally not described as "icons", although "iconic" may be used to describe a static style of devotional image.
  
  After adoption of Christianity as the only permissible Roman state religion under Theodosius I, Christian art began to change not only in quality and sophistication, but also in nature. This was in no small part due to Christians being free for the first time to express their faith openly without persecution from the state, in addition to the faith spreading to the non-poor segments of society. Paintings of martyrs and their feats began to appear, and early writers commented on their lifelike effect, one of the elements a few Christian writers criticized in pagan art — the ability to imitate life. The writers mostly criticized pagan works of art for pointing to false gods, thus encouraging idolatry. Statues in the round were avoided as being too close to the principal artistic focus of pagan cult practices, as they have continued to be (with some small-scale exceptions) throughout the history of Eastern Christianity.
  Nilus of Sinai, in his Letter to Heliodorus Silentiarius, records a miracle in which St. Plato of Ankyra appeared to a Christian in a dream. The Saint was recognized because the young man had often seen his portrait. This recognition of a religious apparition from likeness to an image was also a characteristic of pagan pious accounts of appearances of gods to humans, and was a regular topos in hagiography. One critical recipient of a vision from Saint Demetrius of Thessaloniki apparently specified that the saint resembled the "more ancient" images of him - presumably the seventh century mosaics still in Hagios Demetrios. Another, an African bishop, had been rescued from Arab slavery by a young soldier called Demetrios, who told him to go to his house in Thessaloniki. Having discovered that most young soldiers in the city seemed to be called Demetrios, he gave up and went to the largest church in the city, to find his rescuer on the wall. - Eastern Orthodox and Catholic teaching about Icons


We can see the in the earlier centuries of the Church icons were the ordinary means of Christians expressing themselves in sacred art. The use of statues in the West is a later development in the use of Christian art. 


  The use of icons and symbols in Christian worship
  
  Early Christian art used symbolic and allegorical images mainly, partly no doubt to avoid drawing attention during the persecution of early Christians in the Roman Empire. In the Catacombs of Rome Jesus was represented indirectly by pictogram symbols such as the Ichthys (fish), peacock, Lamb of God, or an anchor (the Labarum or Chi-Rho was a later development). Later, personified symbols were used, including Jonah, whose three days in the belly of the whale pre-figured the interval between Christ's death and Resurrection, Daniel in the lion's den, or Orpheus charming the animals. 
  
  The image of "The Good Shepherd", a beardless youth in pastoral scenes collecting sheep, was the most common of these images, and was probably not understood as a portrait of the historical Jesus. The depiction of Jesus already from the 3rd century included images very similar to what became the traditional image of Jesus, with a longish face and long straight hair. As the Church increased in size and popularity, the need to educate illiterate converts led to the use of pictures which portrayed biblical stories, along with images of saints, angels, prophets, and the Cross (though only portrayed in a bejewelled, glorified state).
  
  After the end of persecution, and the adoption of Christianity by Constantine, large churches were built and from the start decorated with elaborate images of Jesus and saints in mosaic. Small carved reliefs were also found on sarcophagi like the Sarcophagus of Junius Bassus. However large monumental sculpture of religious subjects was not produced, and in Byzantine art and Eastern Orthodox art it is avoided to the current day. It only reappeared in Carolingian art, among peoples who had no memory of pagan religious statues. 
  
  Paintings of Old Testament scenes are found in Jewish catacombs of the same period, and the heavily painted walls of Dura Europos Synagogue in Syria. Catholic and Orthodox historians affirm, on the basis of these archeological finds in the Catacombs, that the veneration of icons and relics had begun well before Constantine I. - Religious images in Christian theology


As we can see, the West started to develop it's own form of tradition with the use of statues after the Church gained Her freedom from the Roman persecutions. Traditions are never still, but are living and always expanding.

The East chose icons and the West chose statues a few centuries later, due to historical events that fell into place.