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

This is a somewhat tricky question to ask because it requires answering why Rome was so much more successful in expanding when compared to the fragmented Greeks. I'd note, first of all, that Roman expansion unfolded across a long span of time, and this theme was present even in its early history, with Rome first expanding into the territories of its immediate neighbors and then slowly extending across Italy.
It should be noted that one of the major advantages Rome had when compared to the individual Greek City States was its much larger population (thanks to a geography far more favorable to farming than what one can find on the Greek mainland). Indeed, when Pyrrhus of Epirus campaigned against Rome, Rome's superior population proved critical: despite Pyrrhus's victories, he was unable to endure his losses he sustained in gaining them (from this term, we get the word, Pyrrhic victory).
Furthermore, be aware that military expansion and conquest proved critical to Roman society, with yearly campaigning an essential feature of the Roman political order. In this way, Rome steadily expanded, initially within central Italy and later across the Mediterranean.
However, as the Republic expanded, it faced increased internal turmoil, which would be further destabilized by the impact of the Marian Reforms, which granted the landless poor access to service in the Roman army. These changes would culminate in the breakdown of the Republic, which is perhaps most vividly expressed in the career of Julius Caesar and his rise to absolute domination of the Roman State before being assassinated in the Senate. Augustus would later complete the transition and bring about a much more autocratic system of governance.
This describes how the Roman Empire gradually evolved, along with the power structures which shaped it. It was created very slowly, and that expansion created significant internal instability (and this instability was something which the transition to Empire could not fully eliminate, as can be observed in the bloody history of the Empire itself). At the same time, we should be aware of Rome's limitations, as far as it's being an imperial power is concerned. In the realm of military administration, for example, distant commanders held the potential to become existential threats to the Emperor (a weakness which left Rome militarily vulnerable as far as the provinces were concerned). Furthermore, be aware that a great deal of local autonomy was actually built into the Roman political structure, with local elites and governors given a great deal of responsibility for seeing to the task of governing themselves.

"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.