In making comparisons and pointing out differences between two narratives, it is important to establish what it is that is being compared and contrasted. For instance, you may wish to analyze the characters and discuss their similarities and differences, or you may wish to analyze themes. Certainly, both stories deal with extra-marital affairs and themes of nature, but the treatment of these affairs differs.
In Chekhov's story, the adulterous affair leads to unhappiness because Gurov is a man who believes,
...in reality everything is beautiful in this world when one reflects: everything except what we think or do ourselves when we forget our human dignity and the higher aims of our existence.
Having abandoned his "higher aims" of the pursuit of love, truth and beauty, Gurov acknowledges that he suffers when he returns to Moscow as he finds the society false as well as his marriage. He has fallen in love with Anna Sergeyevna, and he wants to be "sincere and tender" and respectful of her as he has of no other woman. Anna, too, suffers in their deception; so, they ponder how they will change their lives so that they can begin a new one together.
In contrast to the theme of unhappiness from adultery that exists in "The Lady with the Pet Dog," a passionate instance of adultery is liberating and joyous in "The Storm." When the thunderstorm is over, Calixta watching Alcee depart, lifting her "pretty chin in the air and laugh[ing] aloud." Similarly, Alcee happily writes his wife a solicitous letter, telling her to enjoy her stay in Biloxi a little longer. The passion that he and Calixta have felt has been liberating and enough for them to return to spouses they do love with happier hearts. For these lovers, letting go has brought them fulfillment; for Anna Sergeyevna and Gurov, fulfillment cannot come unless they marry each other.
With another theme, that of pathetic fallacy, the sudden storm of Chopin's story that brews in the skies causes Alcee to seek shelter at Calixta's home, and their passions become as turbulent as the sky. When their passionate act is completed and they are "invited to drowsiness," the rain "beat softly" and then
the sun was turning the glistening green world into a palace of gems.
Calixta, too, beams and turns her face to the horizon as Alcee rides away. In Chekhov's story, while Gurov and Anna Sergeyevna are enjoying their affair at Yalta, the sun shines over the beach, the lovers "admire the sea." As they stare ahead,listening to the waves, Gurov thinks that the constancy of the surf may well be ‘‘a pledge of our eternal salvation.’’ However, once he is back in Moscow, Gurov finds himself uncomfortable. In this chilly, frosty environment, "the places he had seen lost all charm for him." When he encounters Anna at the opera, she is shocked for him. Likewise Anna feels a certain negativism when she first sees him in this snowy environment.
In addition to comparing themes, the characters of each story can be analyzed, demonstrating differences and cmparisons. Certainly, the characters in Chopin's story are more spontaneous and happy in their adultery while those of Chekhov's story become miserable with guilt and chagrin, and they are faced with the dilemma of how to change their marriages.