This is an interesting question as these are not works that are normally juxtaposed. The context for Sonnets from the Portuguese is Victorian England. The sonnets were written in 1845-46 and published in 1850. They reflect a period when women's rights were strictly curtailed. Elizabeth Barrett Browning, especially, did not think her father would ever let her marry. These were also chaste and more religious times, in which women were expected to be pure.
The context of The Great Gatsby is the wild post World War I world of the American jazz age. Prohibition spawned parties where, in defiance of the law, alcohol flowed freely. Women had new rights, such as the vote and new freedom to come and go as they pleased.
As for the texts, Sonnets from the Portuguese is a series of poems meditating on love in personal ways. Gatsby is a different genre as it is a novel about one man's impossible quest to set the clock back and reunite with his first love as if the intervening years had never happened.
The common ground linking the two works is love. In both works, love is a central motivation.
As the eNote I link to notes, the sonnets are divided into three parts. The eNote says:
Helen Cooper, in Elizabeth Barrett Browning: Woman and Artist (1988), divides the poems into three groups: woman seen as the object of a man’s desire and love (Sonnets 1 and 2), the woman struggling to free herself from being objectified and maintaining her own subjectivity (Sonnets 3 through 40), and the woman achieving her own subjectivity while accepting the man’s love (Sonnets 41 through 44).
In the novel Gatsby, Gatsby, the man, also objectifies his love, Daisy Buchanan. By objectify, we mean that rather than treat her as a full human being, he sees her as an object, like a piece of furniture, that he can possess. While he does love her, he loves her in an idealized, objectified way. She represents desire to him: desire for the security her well appointed home in Alabama represented to him when she first met him, desire for wealth, desire for completion, desire to go back in time. No matter what she does for him and how wonderful she is, as Nick Carraway notes, she can never quite live up to Gatsby's dream of her.
Barrett (or the persona she adopts) also idealizes the beloved:
How do I love thee? Let me count the ways. I love thee to the depth and breadth and height/ My soul can reach, when feeling out of sight/ For the ends of Being and ideal Grace./I love thee to the level of everyday's/ Most quiet need, by sun and candlelight.
But as this sonnet, late in the cycle shows, Barrett's love has also matured to accept the here and now, the "everyday ... quiet need." Barrett will say later that she loves "freely." Daisy, in contrast, will never get the opportunity to develop this kind of mature, freely chosen love for Gatsby nor will he for her. He presses her too hard to live up to expectations she can't meet (after all she is married and has a child). Daisy and her husband, Tom, careening carelessly through life, also never seem to stay in one place long enough to develop the quiet maturity that allows Barrett both to accept herself and Browning's love. In the sonnets, love matures, while in Gatsby it is cut short.