This may be more of a linguistic question than a historical question, and I understand that I'm going to be guilty of boiling down a complicated question into a simplistic one. Why is it that we use the word "Greek" (at least in English) to describe both the ancient Greeks and the modern Greeks? It doesn't seem to be a standard we apply to other ancient civilizations -- Romans aren't often called "ancient Italians," unless you're talking about the early days of the Republic and lumping Rome together with the other socii. Modern Greek civilization seems to be significantly influenced by its centuries under Ottoman rule, although it does hold onto its Classical identity very fiercely. And I do understand how important words and names can be -- one just has to look at the anger over the Macedonia/Former Yugoslav Republic of Macedonia naming controversy.
Tl;dr -- Would it be more accurate to use different terminology to describe ancient and modern Greeks (such as "Hellenes" vs. "Greeks"), due to the significant cultural differences?
I cannot answer your question in a solid way. However, the Romans were an empire and a nation, the Greeks were a group of semi-related peoples. While the Romans fell (eventually), greeks never stopped being greeks, even under foreign occupation (rather, it's stuck around BECAUSE it was conquered and went through heavy cultural admixture, not in SPITE of it).