Although they discussed the same issues, and there was some communication between the two, they often approached things differently.
For example, on this page there is a discussion about terumah – the set aside portion for the priests – and ways that it becomes defiled. The Babyloynian school went on to discuss how wheat of terumah which is planted becomes terumah itself. But what if it is defiled?
Abaye b. Abin and R. Hanania b. Abin studied (the Tractate) Terumoth at Rabbah's academy. Rabbah b. Mattenah met them [and] asked them, What have you discussed in Terumoth, at the Master's academy? — Said they to him, But what is your difficulty? He replied. We learned: Plants of terumah which were defiled, and he [their owner] replanted them, are clean in that they do not defile [other eatables], but they are forbidden to be eaten [as terumah]. But since they are clean in that they do not defile, why are they forbidden to be eaten?
The analysis continues, but their conversation that defilement is solidified by the priests “mental neglect;” that is, he put it out of his mind and thus it is no longer terumah.
Thus did R. Shesheth say, he answered, what does ‘forbidden’ mean? They are forbidden to priests, since they became unfit [for eating] through [his] mental neglect.
The rabbis back in Palestine were, shall we say, not impressed:
When Rabin went up (to Palestine from Babylonia) he reported this teaching with reference to the terumah plants (R. Shesheth’s statement) before R. Jeremiah, whereupon he observed: The Babylonians are fools. Because they dwell in a land of darkness they engage in dark [obscure] discussions.
Nice comment – maybe having to do with the fire-worshipers of Babylon who forbade Jews to use fire during certain Babylonian holidays. Or maybe it’s just a commentary on how easy it is to get lost in obscure discussion and forget the light.