The most obvious difference is the microphone polar pattern. The SM 58 has a cardioid polar pickup pattern, while the Beta 58A is a supercardioid. The polar pattern subsequently affects how you should use the microphone and where it would be most appropriate.
As you can see from the diagrams, a cardioid microphone will reject best from the back while a supercardioid microphone will reject better from the sides. Both microphones pick up sound best from the front, but a supercardioid polar pattern has the added advantage of being more directional and therefore less susceptible to feedback when using correctly placed stage monitors.