Why did Rome's conquest of Carthage lead naturally to conflict with the Hellenistic Kingdoms?
This is a very intelligent question. Right after the war with Carthage, Rome entered into conflict with the Greeks. The question is why. What exacerbates this question is the Romans suffered a great deal in three major battles with Hannibal: Trasimemne, Trebia, and Cannae.
The best explanation is that Rome was imperialistic. Let me make two points in connection to this.
First, the Romans needed to conquer to keep their economy and way of life going. The best place to gain riches was the East. The Hellenistic East was fabulously wealthy, not to mention cultured. So, if the Romans could conquer under the guise of setting the Greeks free (think of the consul, Flamininus, and the Isthmian Games where he pronounced the freedom of the Greeks), they would gain in two ways: material wealth and social prestige.
Second, the whole Roman constitution was built on competition. Two consuls who are elected yearly can only lead to competition among the elite. Moreover, the best way to distinguish yourself was through military conquest. What was left? The East.