There's a bunch of reasons for this.

Hannibal was in enemy territory, had no siege equipment, could not get provisions to starve Rome out without starving his own men, and needed to be very careful all the time - one slip up and he was done for.

Alaric was able to just storm the city as it had almost no defenders, he commanded the largest force in Italy and the Romans simply lacked the resources to fight him off. At the time of Alaric, most of Rome's forces were fighting in other parts of the empire, and the ones in Italy were mostly concentrated around Ravenna (which was then the capital instead of Rome).

Mainly, in Hannibal's time, the entire Republic was busy fighting Carthage. Hannibal could not be resupplied, because the Roman navy controlled the seas. Outside of Italy, massive Roman armies were attacking Carthaginian lands.

Meanwhile, in Alaric's time, Alaric was just the umpteenth crisis at the same time. Gothic lands were not threatened by other Roman armies, since the Goths had no land. There were no issues with supplying Alaric's army, since he could live off the Italian land without fear of guerilla warfare (as happened to Hannibal, which prevented him from sitting down in one place too long).

In short, Hannibal and Alaric fought completely different versions of Rome. One was unified, focused and patient, and one was divided, distracted and badly lead by one of Rome's worst emperors.