This is a complicated assignment as there doesn't seem to be an overarching rationale for choosing these two works for comparison. There are, however, a few areas it might be possible to explore.
Lord Acton famously said:
Power tends to corrupt and absolute power corrupts absolutely. Great men are almost...
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always bad men, even when they exercise influence and not authority; still more when you superadd the tendency of the certainty of corruption by authority.
One central theme of both texts is how power corrupts those who wield it.
In Shakespeare's play, Prospero has been deposed unfairly by Antonio with the connivance of Alonso. Not only do these two wield power unjustly, but so does Prospero, who himself has deposed Caliban, the rightful owner of the island, and enslaved Ariel. Prospero himself uses his magical power despotically, though he reforms by the end of the play. Similarly, two types of misuse of wealth and power are seen in Fitzagerald's novel, with the Buchanan family show abuse of hereditary position and wealth and Gatsby the corruption of the new wealth derived from alcohol during Prohibition.
Another theme you could address is the power of love. In both stories, it gives characters a type of strength that drives them to achieve deeds that would otherwise be beyond their normal range of action. In the case of Ferdinand and Miranda, though, this is seen as a positive power and in the case of Fitzgerald's characters it is far more morally ambivalent, leading as much to ruin as redemption.
Another major theme is social stratification. One could argue that in both works, the upper classes oppress the lower in a way that morally corrupts both.
In terms of themes, there are a number of options. You might, for example, compare how each text explores the theme of love. Gatsby's love for Daisy is arguably a positive influence on his life inasmuch as it drives him to make, at least ostensibly, a success of his life. You might also argue, however, that his love for Daisy is obsessive and unhealthy. In The Tempest, there are similar points to be made about Prospero's love for his daughter, Miranda. This is of course a different type of love than that which Gatsby has for Daisy, but nonetheless, like Gatsby, Prospero's love is the main driving force in his story. It is also, like Gatsby's love for Daisy, perhaps an obsessive and destructive love.
A second theme you might compare is the theme of power. Tom Buchanan, in The Great Gatsby, exerts a physical power over Myrtle, which culminates when he hits her for mentioning his wife's name. In The Tempest, Prospero uses his magic powers to control all of the other characters on the island, much as a puppeteer might control puppets on a stage. In both texts, there is an argument to be made that too much power inevitably leads to abuse and suffering.
A third theme present in both texts is the theme of tragedy. At the end of The Great Gatsby, the eponymous Gatsby meets a tragic end at the hands of George Wilson. In The Tempest, although the ending is happy, there is much tragedy in the rest of the play. Caliban and Ariel's lives as slaves are arguably tragic, as is the life of Alonso while Prospero has him believe that his son, Ferdinand, is dead.
In terms of the style of both texts, you might compare how effectively Fitzgerald is able to communicate the aforementioned themes in prose with how effectively Shakespeare is able to do so with stagecraft. It might seem advantageous, for example, for a playwright to be able to utilize actors, costumes, and props to communicate to an audience more directly and perhaps more visually than a prose writer is able to do. On the other hand, perhaps a prose writer has greater access to the limitless scope of the reader's imagination and so, arguably, can explore nuances of meaning more easily than can a playwright.
There are lots of other options, too, in terms of style. You might, for example, compare the dialogue in one text with the dialogue in the other, or the use of settings or symbols in one text with the use of settings and symbols in the other.