I would say that much of the key to this answer can be found in the geographical situation of each of these ancient powers.
Sparta is located in the heart of a very mountainous and rugged part of southern Greece. Because their home city is relatively locked in by the mountains, the Spartans would have great difficulty in accessing other parts of Greece. The Spartans did have access to the sea, but that access was also not very close by. Finally, the Spartans relied on an unstable population of slaves, known as the helots, to do much of the menial labor that allowed the Spartan males to devote their lives to warfare. Unfortunately for the Spartans, any time that the helots had an opportunity to revolt, they did. Thus, the Spartans could not turn their backs on the helots for a moment.
The Romans' capital city, on the other hand, is very close to the Mediterranean coast. Thus, the Romans had rather easy access to the sea, over which they could send their governors and military personnel in a relatively rapid fashion. Futhermore, Italy is centrally located in the Mediterranean, so that the Romans were in an excellent position to reach out to Spain, France, Greece, and northern Africa.
The Romans were also excellent engineers and they built roads all over their empire, which gave them good access over the land to the other parts of their territory.
So, I would say that it was not that the Roman political system was that much better than that of the Spartans, but that their geographical position in Italy made it far easier for the Romans to control other peoples than it was for the Spartans to control others.
One further note...in terms of population, the Romans had infinitely superior numerical resources than the Spartans ever dreamed of having. Even today, Sparta is a very small town, whereas Rome remains on the world's largest cities. The key to much of this: location, location, location!