Gatsby and Dimmesdale are similar in these ways.
They are both private people who create public images that are very different from their real natures. Both exercise power; Gatsby wields economic power, while Dimmesdale has spiritual influence over his community.
Both die as a result of decisions they make. In this regard, each man controlled his own fate.
Both are capable of great romantic love. Gatsby clings to his love for Daisy throughout his life even to his death, creating and living a destructive version of himself that would be acceptable to her. Dimmesdale continues to love Hester. After years of being separated from her, he meets her in the forest and makes plans to run away with her, despite the fact that this violates his spiritual life.
Their differences are numerous, also.
Gatsby is quite worldly; Dimmesdale is a man of the spirit. Gatsby gives no thought to the morality of his acts; Dimmesdale obsesses about his own immorality.
Although they both die, their deaths are significantly different. Gatsby dies alone in his pool, waiting for Daisy to call. Dimmesdale dies in Hester's arms, reunited with her and with Pearl. Unlike Gatsby, Dimmesdale dies in peace.
Gatsby never faces reality in regard to his relationship with Daisy. He makes excuses for behavior until the end. Dimmesdale recognizes that he and Hester can never be together in this life. He abandons their plan to run away together and chooses to publically climb the scaffold instead.
Characteristics: Both Gatsby and Dimmesdale are mysterious characters. Gatsby is presented to the reader as a man full of mystery as other characters in the book discuss rumors of his lifestyle. Dimmesdale is also presented in the same way; although the characters in the novel are not discussing rumors they have heard about him, they do not know much about his life behind the scenes -- his secret fasting and self mutilation and the fact that he is the father of Pearl. For this reason, both characters have the ability to get away with many things that they could not get away with if their backgrounds were revealed to all. The major difference in their characteristics deals with their motives -- Gatsby is motivated by his lust for Daisy while Dimmesdale is motivated by the guilt that he feels for what he has done.
Their Fates: The similarities in the fates of these two men could lie in the final outcome -- both men meet their tragic end and die in the end of the novel seemingly because of the women whom they loved. The differences in their fates is that Gatsby never fully achieves his dream of getting Daisy yet Dimmesdale is successful in finally releasing his bottled up guilt when he reveals his sin to the people of Boston.