This is a great question. There are several reasons why the Roman army became great in numbers and not the Greek city-states.
First, as you ask in your question, the Greeks did not develop an empire. They had city-states. So, these cities were much smaller than the Roman Empire. Let me give you a sense of scale. Plato believed that the perfect sized Greek city was comprised of five thousand men. This is tiny compared the the Roman Empire. So, the most important point is that we dealing with a different scale.
Second, the Romans assimlated people, whereas the Greeks did not. If you read even the first chapters of Livy, one of the great Roman historians, you will see that the Romans assimilated the Sabines, the Etruscans, and other people fairly easily. This swelled their numbers and land in time.
These two reasons are the most important and make all the difference.