To get at the core of this question, a comparison between Robinson's characters and Anderson's, let's examine one poem each: "Richard Cory" (Robinson) and "American Spring Song" (Anderson).
Even a shallow reading of these poems will reveal a vast difference in the kind of characters each of the poets presents. Robinson's character, Richard Cory, is "a gentleman from sole to crown"; he is "quietly arrayed" and "human." The people in the town Cory comes to visit wish they were him. And yet, the end of the poem is shocking in its brevity. Cory returned home and "put a bullet through his head."
This seems to be the story of a man who is totally unsatisfied with his life. Despite the fact that Cory is "richer than a king," he kills himself. His dissatisfaction is in horrible contrast to the townsfolk who all live very meager lives—no meat, only bread. They continue their work and their lives, waiting for the light. Cory has no reason, in the context of the poem, to be unhappy with his life. Indeed, he should celebrate his fortune all the time.
Cory is a flawed character. He seems to be a decent chap throughout the whole poem, until his deep (unwarranted) dissatisfaction is revealed. We can all see some of ourselves in him—none of us are perfect.
Contrast this to Anderson's grotesque narrator in "American Spring Song." This character, one spring morning, decides to be "glad because of [his] brutality." Right away he is shown to be different from Richard Cory, who hides his sadness, stuffing it down until it finally bursts. No, the narrator in "American Spring Song" revels in the violence he is about to inflict. He punches the men and women of his town until his "hands [begin] to bleed."
The narrator sits at the river's edge and, in the distorted reflection of the water, sees himself a "grotesque little god with a twisted face / A god for myself and my men." Again, this is a huge leap from Richard Cory, who is nothing if not human, as he is flawed and imperfect like all humans.
In Anderson's poem, the narrator says that his (self-diagnosed) insanity comes from being "long alone in a strange place where no gods came." In his ravings he cries, "Creep, men, and kiss the twisted face of my mud god . . . I'm a twisted God myself."
Both Robinson and Anderson are aiming to reflect the human experience. Robinson does so by presenting humans who seem good but hide a hidden flaw or sadness. Anderson does so by extreme hyperbole, asking us to find ourselves in the twisted, violent man on the mud bank.