With a question like this, it is important to keep in mind that the modern nation of Greece exists as a unified country, comprising the Greek peninsula and the Greek islands in the Aegean. This observation does not really apply to the Ancient Greeks, who did not possess the same sense of national unity, divided as they were into various city-states.
The one key element to keep in mind is that Classical Greek civilization was very far flung, spreading across the larger Mediterranean. On the one hand, it would have comprised the territories currently recognized as Modern Greece (in the Greek mainland and the Aegean), but its extent stretched much further. For example, you can find Greek city states and colonies present in such places as modern day Turkey, as well as in southern Italy, North Africa, and even in countries further west such as France (the city of Marseilles, for example, began as a Greek colony) and Spain.
In this context, Ancient Greek culture had a profound influence on world culture and history. Perhaps the most famous and significant influence can be observed in the Romans (Ancient Greece had a strong cultural influence on Ancient Rome, an influence that can be seen reflected in such fields as literature, arts, religion, architecture, etc). In addition, one can also point towards the Persian Empire, with its history of conflict and rivalry with the Greeks.