Look at the comparison between the two. A storm is a large, widespread disturbance. A teacup is a very small, confined space.
To describe something as a storm in a teacup would be to say that the level of activity is is way out of proportion to the prevailing conditions. Consider also that the intensity of a storm would be that much more amplified if it was compressed into such a small space.
Essentially, to make a major fuss over what would otherwise be a minor issue.
All that being said, while "storm in a teacup" may be in use somewhere by someone, the more common form is "tempest in a teacup." We like alliteratives.