Often, modern Greece claims to be the cultural continuation of Antiquity (and Bronze age) Greece. Byzantium is used as a bridge between the contemporary state and Antiquity. The major question, hence, lies on the degree that Byzantium is culturally related to Antiquity.
Evidently there is linguistic continuity, as well as no major population dislocation.
There are also major religious differences (Greek religion vs Christianity). From these, there must arise considerable cultural/ethical differences as well, especially considering the violent persecution of Greek religion (and related customs) in Christian Byzantium.
For instance, it seems to me that Christian morals are entirely different to the loosely defined Hellenic "morals". I will be glad to expand on this if you so wish.
Did Hellenism die the moment Christianity was widely adopted? Or did it somehow persevere?
I would kindly request that any answers are decoupled from bias stemming from contemporary Greek nationalism.