I think that one point of similarity between both narratives is that the female protagonist in each is discarded, unwanted by some element of the prevailing social order. Shakuntala is discarded by Vishwamitra and abandoned in the wilderness. Cinderella is discarded by the stepmother and her stepsisters. Another similarity is that both must endure some level of pain and forlornness in order to find redemption. Shakuntala must endure not being recognized by her love, King Dushyanta, and lives in self- imposed exile because of it. Cinderella is constantly the target of scorn and abuse until she is granted the wishes to attend the ball and is rescued by the prince. Along with both featuring loves of royalty, there is a redemptive notion at the end of each story.
The differences exist, as well. One of them would be plainly that Shakuntala's narrative might be more painful than Cinderella's. While Cinderella did have some difficulty in her experiences, I find Shakuntala's falling in love only to find, through really no fault of her own, that her lover fails to recognize her. As a result of this, she goes deep into the forest, gives birth to Bharata and lives on her own. Cinderella's love is not recognized until the end, while Shakuntala must live with both the pain of loving and losing one's love. Another similarity would be that there is a punitive and restorative element to the moral order at the end of the Brothers Grimm tale in that the doves blind the sisters, whereas in Kalidasa's narrative no such punishment is present. The Rishi Durvasa is not punished for his fairly harsh curse on Shakuntala, and the ending is one of reconciliation without the punitive streak of external justice present.