How did the ancient Romans create a huge and long-lasting empire, when the ancient Greeks were never able to unify for very long?

The most important factor was geography. Greece is extremely mountainous, making land travel between different regions quite difficult. Although Greece had been inhabited in the neolithic age, Greek culture as we think of it began in a period know as Early Helladic III, starting in roughly 2200 BC as Greek-speaking peoples gradually moved into the area that became Greece. These Bronze Age towns grew in a pattern that was determined by geography. Usually there was an urban center, often a fortified acropolis, consisting of civic, religious, and domestic structures surrounded by farmlands and a harbor. These functioned as city-states (polis; plural poleis). External trade and communication was usually by sea, and cities were as likely to trade with islands, Asia Minor, Sicily, or Egypt as with each other. Because city states could not expand in their own regions (due to the surrounding mountains), they planted colonies in places like Italy, North Africa, and the east coast of the Mediterranean. Colonies usually remained loyal to their parent cities. Because of this early history, ancient Greece simply did not exist as a nation. The city states might share a language in common and form alliances, but they did not, until the Hellenistic period, consider themselves part of a common political unit and thus did not form empires, as they did not individually have the population or wealth necessary to do so.
Rome, on the other hand, existed as a unified state, gradually expanding through Italy and then the Mediterranean.