There has been a lot of arguments in modern revisionism questioning if Caligula was indeed insane or if it was a fabrication of contemporaries that disliked his populist and demagogic attitude.
His reign was particularly uneventful in the grand scheme of things, unlike those of... well every other Emperor immediately before and after him (notably Claudius and Nero), but one thing I consider notable is that he was the first Emperor assassinated by the Praetorian Guard, and this set a very dangerous precedent in the history of the Empire. In fact, it is possible that his "insanity" might have been exaggerated exactly to legitimize the coup by the Praetorians and the ascension of Claudius to power.
If we avoid the Praetorian coup, we can suppose that the Julian Dynasty might survive a bit tad longer. I'm not sure what consequences this would have, but if we have a dynasty with at least a veneer of legitimacy - associated with the Augustan heritage and well connected to the actual Italian Roman aristocracy, instead of provincial origins - , we can avoid the constant cases of usurpation and civil war, though this depends on a lot of butterflies.