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.