Why was the Roman Empire was more centralized than classical Greece?

"Classical Greece" was really not a single nation or state in the modern sense. Instead, the Greek mainland and nearby islands were a series of independent city-states (poleis) inhabited by people speaking a common language, albeit with dialectal variations. One significant reason for this was geographical. Greece is mountainous, making land travel between different Greek cities quite slow and difficult, something that would have impeded both political unification and military conquest. We see a similar pattern in the history of Switzerland, also a country geographically divided by mountainous terrain. Greece was not actually unified in the classical period until conquered from the outside by Macedon. In a sense though, individual poleis such as Sparta and Athens were strongly centralized, with the city controlling the surrounding countryside; it is not that the ancient Greek city states were not centralized so much as that they remained small, centralized, tightly knit communities. The degree of independence of colonies was in part due to colonies being considered nascent independent states.

Italy itself was gradually unified mainly by the pressure of external conflicts. The Romans ran into conflicts with the Etruscans quite early, and later had a series of wars with the Carthaginians. As Rome almost accidentally developed an empire through success in various wars, it needed a way to administer that empire. In part due to the need to award land as an incentive for military service and in part due to the need for control over the Egyptian grain supply, Rome developed a more centralized imperial bureaucracy, although daily administration of its far flung colonies was often left to provincial aristocrats.