The same principles apply as when you normally want the differential form of something vs. the integral form.
It's like asking when do you want to know the total distance and time of a road trip vs. what the speed was at a particular point.
With a uniformly charged sphere, for example, you can use the integral form to get the total flux at some radius, and then use that to infer what the potential is at any point because of the perfect symmetry of a sphere. Wouldn't work with a cube!
The analogy would be if you know you drove the exact same speed throughout an entire road trip, you could calculate what the speed was by dividing total distance / total time. Not so if there was a lot of starting and stopping!
