"The key point is that the potential, rather than the field, has an observable effect. In classical gauge theories, the potential itself is not considered to be observable; only the field (the gradient of the potential) is. However, in QM, the potential itself can have observable effects."That comment is short, succinct and understandable, it's the best summary of the physics involved that I've heard in so few words.
It seems to me that we don't put sufficient emphasis into considering potentials. Viewing things from the aspect of potentials I reckon changes one's perspective, and I think the Aharonov-Bohm effect is a case in point, without doing so we easily come aground.
However, the Aharonov-Bohm effect still perplexes me more than I'd like - in QM, why exactly does potential have observable effects? Why is it that a phase shift occurs in the wave function of a charged particle in the near vicinity of a solenoid even though both magnetic and electric fields are negligible? (I'm either not convinced by explanations that I've heard to date or thst I don't fully understand them.)
Whilst my understanding of the Aharonov-Bohm effect is deficient in this regard, it nevertheless seems to me that if the effect can also be detected in a gravitational field context then we've likely found a profound and deeply relevant connection between EM fields/relatively and gravity. If verified, then we'd have to consider it a breakthrough in our understanding of what up until now has been an intractable problem.
The key point is that we've now experimental evidence for such a connection and that's really good news.