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In one sense, it is a shame, because if it wasn't for the war, they might have been good friends. They both despised organised religion (although they were not atheists) and Franklin's admiration for Gibbon (who is regarded as the father of modern history because of his innovative concept of using primary sources as much as possible instead of second and third hand accounts) meant that these two intellectual giants of their times could have had many meaningful discussions.

However, Gibbon was a constitutional monarchist by conviction, and Franklin was a republican. When I read Gibbon's first volume, his justification for monarchy made a lot of sense (for the time). He partly ascribed the collapse of the Roman Republic on democracy, which to his horror, seemed to be being advocated by the American colonists, and blamed giving the 'mob' legislative power for the downfall of the Roman republic and the rise of the hypocritical despotism of the Roman Principate. As far as he was concerned, the best form of government was a British style hereditary monarchy tempered by an independent legislature that controlled the purse strings. The hereditary monarchy ensured continuity and a peaceful transfer of power, whilst the legislature (parliament) ensured that in the event of a bad monarch residing on the throne, the worst excesses could be avoided by the legislature refusing to grant money to support the monarch's crazier schemes, as well as threatening to do so if the King tried to turn his kingdom into a tyranny. He also would have been aware of the pitfalls of getting rid of the lawful king given what had happened with England's brief republican experiment in the previous century (which had led to Cromwell's military dictatorship).
He viewed the kind of government Franklin advocated, rightly or wrongly, as a recipe for disaster that would eventually lead to the collapse of order and the rise of a military tyranny, which as well as being oppressive, would also be unstable as army commanders fought with each other to become emperor as the did during the third century in the aftermath of Alexander Severus' assassination. I don't think that the colonists attempts to model themselves partly after Rome inspired confidence in Gibbon, given what he knew about the fate of Rome from his extensive research.
Franklin for his part wasn't very forgiving of those who took the opposite view to his own considering he disowned his own son for siding with the Tory cause. If it wasn't for the war which had broken out which was in many respects an ideological conflict as well as a secessionist one, these debates between the too men could have remained as a source for amusing coffee table debates between the too men, but in light of the fact that the two sides where at war with each other and spilling blood, I don't think it was possible for the two men to converse with each other about their two respective positions without their being bitterness and enmity between them...