It is clear that the biggest relationship between the plot of this excellent short story and the "local colour," as you put it, is the way that the storm is actually a symbol of the far bigger and more dangerous storm of passion that occurs between Calixta and Alcee during the story. Note how their passionate union occurs during the storm:
They did not heed the crashing torrents, and the roar of the elements made her laugh as she lay in his arms. She was a revelation in that dim, mysterious chamber; as white as the couch she lay upon.
Under the cover of the storm, they are both free to indulge in a long-standing infatuation and finally satisfy their desires, in spite of their mutual marriages. The very last line indicates the importance of the storm as a symbol for the entire short story:
So the storm passed and every one was happy.
Having given free reign to their desires, both Alcee and Calixta appear to be able to return and enjoy their marriages. Thus the local colour of the storm and the weather is inextricably intertwined with the desires of the two central characters and how they meet those desires.