It seems that the more ancient Greek would be more likely to die out than Latin. Why would Latin change into Italian (French and Spanish I can understand, but why would it get lost even in its homeland?) while Greek persists?
The simple answer to this is that someone in modern Greece can no more understand Aristotle than an Italian can freely read Lucretius, or you can just pick up and read Beowulf. It's still called "Greek", but they're not the same.
Modern Greek has about as much in common with ancient Greek as Spanish or Italian does to Latin. Both have changed profoundly over the centuries. If you want to be extra technical about it, Italian and the other Romance languages are just dialects of Latin, just as Greek is to Greek.
No one maintained a distinctly Latin tradition. As you pointed out, it became Italian, French, Spanish, ect, while there was certainly someone bothering to maintain Greek.
The thing people tend to forget is that any language spoken 500 years ago is fairly distinct from the same regional language(s) spoken today. We call it "old English" for a reason. Similarly what the ancient Greeks spoke probably isn't entirely Greek to modern Greeks. If that makes any sense.
Language is fluid. It's always changing.