To write a compare and contrast essay about these two works, first you need to find a thesis focused on what they share in common. As they are works from different periods in different genres, your choice must be thematic. The only really common theme here is that of love, and the way in which it can serve as both a creative and a destructive force. Once you have introduced your theme, you should outline key areas of contrast:
Genre: Sonnets from the Portuguese is a sonnet sequence and The Great Gatsby is a novel. As the sonnet is a verse form traditionally associated with an intensely personal account of love, you can discuss how Browning deals more deeply with internal feelings and Fitzgerald more with social context.
Period: Browning wrote in the nineteenth century as a woman living very much under the control of a domineering father. For her, love represented both an escape from one form of patriarchy and the potential of moving under a different form of patriarchy which, although more pleasant, might equally subsume her as writer and person. You could contrast this with the greater freedom enjoyed by Daisy and Jordan, but also discuss how they are also products of their environment with its own restrictive conventions.
Narrator: Gatsby's narrator Nick stands outside the romance between Gatsby and Daisy while Browning writes as a first person protagonist. On a textual level, you can look at how these narrative choices provide the reader with a different emotional experience.