The big disadvantage I see with Shot Spotter is that it does not tell you whether someone was hit. At least when someone calls 911, the dispatcher can ask them if they are aware of someone being injured. If the caller doesn't know, they can send a patrol car when available.
If the caller says there is a guy bleeding on the ground, they can dispatch police and EMS urgently.
Shot Spotter seems to tell police departments with certainty that those were gun shots, and puts a (poorly defined) obligation on the police to respond.
As an advantage, Shot Spotter seems like it could save police resources regarding people who call thinking they heard gunshots, when instead they heard merely fireworks. In this latter instance, they would have the info to assign the call a very low priority, and attend to more urgent matters. At present, the dispatcher is forced to decide whether the caller is credible re: knowing the difference between gun shots and other sounds.
...I suspect that (at present) dispatchers are coached on what makes a call credible, and those who repeatedly send the police on erroneous urgent calls are unlikely to remain in their job for long.