First, you have to keep in mind, that Rome in 390BC is not the same as Rome in the late 3rd century BC. When the Gauls sacked Rome it neither controlled all of Italy nor did it have a strong wall surrounding the town. It also was a single event, long before genuine Roman historiography began, while the 3 punic wars were spread over more than 100 years in an era, where the Romans themselves were producing literature, including historiography. You can assume it was just much more present in Roman life. You also can't neglect the aspect, that Rome began to gain dominance and supremacy over the mediterranean area after defeating Hannibal. That being said, the Romans did indeed remember the day Rome was sacked by the Gauls in shock and sadness. It even became a dies ater, a black day, in the Roman calendar.

But whatever the reason, the impression that Hannibal, or rather Carthage, was the greatest threat to Rome is not modern. In augustean time for example, you can see the importance of Carthage especially in Vergil's Aeneid, where the clash of Rome and Carthage is foreshadowed in mythical context. Further, if you read Seneca's epistula 51, you can conclude, that Hannibal was seen as a prototype of a grim, determined, hard person, although Seneca uses him to show the danger coming from a nice, warm place with too much luxury, when Hannibal, who even subjugated the Alps, is beaten by that place, because it made him soft and weak.

I am sure, there are even more (and better) sources for the Roman reception of Hannibal, but they don't cross my mind right now. I hope this helped though.