Copying and preserving, but not really advancing? It seems to be leveled that Byzantine philosopher were mediocre.
As to be expected whenever you live in a poor realm. A Byzantine Empire that survives in the Middle East would be much wealthier realm with larger and more well developed cities, which could easily create the kind of middle classes that pursue intellectual development. As it was, the Byzantines were quite poor. Urban centers in Anatolia and the Balkans drastically shrunk as food supplies from Syria and Egypt dried up, with Arab raids causing further economic destruction.
The Muslim tradition of science and philosophy developed under the conditions of a vast, wealthy empire, it didn't just crop up because of the Koran or anything silly like that.
As a further point, although it's about as cliche as anything else, many leading Byzantine scholars have pointed out the dearth of sources. It's possible Byzantine intellectuals came up with various concepts at similar times to their Muslim rivals, and probably received knowledge from them from the Muslims themselves at any rate. We just don't know enough. We have partial copies of many texts made by Emperors themselves, so it's not too surprising that we see very little evidence of the Byzantines preserving or advancing science and philosophy (or at the very least them gaining these advances from their Muslim neighbors.)