What caused scientific and cultural decline of East Roman (Byzantine) Empire?

The Byzantine empire did not really "decline" per se, it was degenerate from the beginning.

The entire Greek world in Roman times was very rich and decadent. Greeks with any talent generally moved to Rome or Alexandria to do their work. When Rome collapsed the Greeks kept on doing what they were good at: being rich and decadent. Very little art or science was produced in Byzantium. It was mostly a trade hub. The surviving works we have produced in Byzantium tend to be degenerate scholia and Christian tracts. Most of their "scientific" enquiries were centered on determining whether certain beliefs were "heresies" or not. The only Byzantine scholar I have ever found to have produced useful information is George Syncellus. Reading Syncellus is very funny because every time he quotes old sources he makes sure to describe them as "demonic" and their writings as the "work of the devil".

The question itself needs to be addressed more comprehensively and accurately.

First, the Byzantine Empire, was a 1000 year old Empire that had one of the few Universities and Libraries on the European continent during the so-called, "Dark Ages". The Library and University of Constantinople was founded in the early 400's and was destroyed by Latin Christian Crusaders in the early 1200's-(whereby the city of Constantinople, subsequently, came under Latin/Papal rule for much of the 13th century). However, during much of Byzantium's imperial heyday, many of the Classical Greek works WERE meticulously preserved, commented on and taught by the Ecclesiastical Hierarchs. (One can still find Medieval Greek translations of Hippocrates, Plato and Aristotle).

Second: While the Byzantines meticulously preserved much of their Hellenic intellectual heritage at places, such as, The Library and University of Constantinople, as well as Saint Catherine's Monastic Library in the Sinai-(as well as in other Monasteries throughout the Byzantine world), yes, when compared with the intellectual advancements made by the Muslims-(both Arab and non-Arab) during the Medieval period (and even some of the Astronomical achievements made by the distant Mayan civilization in Mexico and Guatemala), the Byzantines did lag behind and were, comparatively and intellectually speaking....somewhat static. It is difficult to find an archeological wealth of Byzantine Scientists, Mathematicians, Physicians and creatively minded Luminaries-(similar to their Ancient Forefathers). Admittedly, Constantinople, was not Ancient Athens, nor was it Ancient Alexandria, nor was it Ancient Ephesus, nor was it Medieval Baghdad, Cairo, Fes, Cordoba or Toledo. Byzantium, was an important cultural Center, but not known for its intellectual advancements and ingenuities in the above mentioned fields.

Third: However, in addition to its unique Ecclesiastical architecture, the Byzantines did make other cultural achievements in iconography, hymnal music, as well as helping to revive the writings of Plato-(within a strict Christian theological perspective), as well as helping to construct some well built and fortified Castles and City walls-(both in Constantinople and elsewhere in the Empire). But, in addition to these artistic and urban planning achievements, the Byzantines-(especially during the early years of the Empire), produced major Theological writings and commentaries, more commonly known as, "The Church Fathers....of the East"-(acutely distinguishing itself from the Roman Catholic Church Fathers in the West). The earliest origins of the Slavonic languages were co-pioneered and co-developed by 2 Greco-Thessalonian Saints, Cyril and Methodius during the 800's.

Fourth: But, perhaps the main cultural achievement or legacy among the Byzantines, was their steadfast and largely uninterrupted ability to maintain the Greek language throughout Constantinople, Greece proper and much of Asia Minor in a variety of functions,. ranging from the everyday Speaker, to the University Professor, to the Diplomat, to the Merchant class and above all....to the Ecclesiastical and Imperial Aristocracy. It is perhaps this sole achievement that is the most significant within the cultural legacy and identity of Byzantium.

ERE in fact did significantly contribute to science and culture, especially law, building, military and philosophy. I encourage you to read this discussion on reddit for more details: http://www.reddit.com/r/AskHistorians/comments/1k1jso/the_byzantine_empire_often_gets_remarkably_little/

There wasn't necessarily a "cultural decline" in the Byzantine Empire. It depends on what you mean by, "cultural decline". If you are comparing it to the intellectual creativity and originality of Ancient Greece-(from Homer, to the Alexandrian Scientists of late antiquity), then yes, the Byzantine Empire would appear to be culturally stagnant and unoriginal.

However, if you properly contextualize the Byzantine era as its own independent civilization, then you would have to take into consideration many original and distinct cultural achievements, including, iconography, architecture, music, the establishment of monasteries and seminaries, the founding of the University of Constantinople and its famed Library-(rising from the ashes of the destroyed Alexandrian Library and University, as well as following the closing of Plato's Academy), the meticulous preservation and translation of the Greco-Roman "classics", Linguistics-(i.e. the establishment of the Cyrillic alphabet by the Greco-Thessalonian Saint Cyril), advancements in military technology-(i.e. "Greek Fire"), as well as ship building and producing generations of Church Fathers and Theologians during the Empire'e early years, including, Saints Athanasius, Basil, Chrysostom-(and many others). As you can see, it was hardly a "cultural decline".

Admittedly, the Byzantine era was no match for the ingenuity and originality of Ancient Greece, yet at the same time, the Byzantine civilization was never part of the larger European "Dark Ages".