Classical Greece is named mainly after Athens spread Greek culture around the world. Classical period lasts for about two centuries, after Persian Wars till the death of Alexander the Great, were Greek civilization transforms the global civilization as Hellenistic.
Classical Athens and generally Greek, is about democracy, theater & drama, architecture, mathematics, philosophy, and everything that was taught in ancient Athens and Alexandria, which were the main centers of science. Classical world continued until 529 AD and the closing of Athens schools.
Thankfully, today 17 foreign schools are based in Athens to study Classical period. These schools offer a lot to modern archeological science and they produce many events in Athens and many of them are located around Plaka and its streets.
If today someone wants to visit Classical Greece, he should start from Athens, get to Ancient Olympia, Delphi and finally Delos. Aegean is related to earlier Cycladic civilization and Crete to later Minoan. Modern Athens was built as a european classical thematic park, that bear neoclassicism, something that is obvious with a walk around the city.
Classicism is mostly a concept, an idea, something general and abstract sometimes that described human development is scientific framework. But you have to come to Athens to feel it!