Ironically, Richard Wright's story "The Man Who Lived Underground" (1944) is a condensed version of a novel he wrote but could not get published, and Ralph Ellison's novel Invisible Man (1952) sprung from a successful short story "The Battle Royal" that he published.  Regardless, both are descended from Fyodor Dostoyevsky's modern novella Notes From Underground (1864), which chronicles a sick man's withdrawal from Russian society.  

As African-Americans  writing in pre-Cold War America, both Wright and Ellison were lured to the Communist Party, which championed racial integration and even funded their artistic efforts.  Both Wright's story and the section of Ellison's novel involving the Brotherhood deal with how the Communist Party exploited the individual for the sake of the many.  Critics have said that Wright, much more involved in the party, was beginning to extricate himself from Communist dogma at the time the story was written.  Like the invisible man (narrator), Ellison was lured to the party, but he never fully took up its fight.  The plight of Tod Clifton, in particular, who becomes involved in the Brotherhood (Communism), then forsakes it for Black Nationalism, only to be shot by a white policeman, closely parallels the actions of Fred Daniels (who also is shot by a white officer).

The circumstances which drive Fred Daniels, the invisible man, and the underground man are not primarily race-related or even culture-related.  Rather, they are human related.  Their decisions to exist on the periphery of society is existential, not a black- or Russian- or male-related problem.  Individuals decide to go rogue, move underground, or change their identities in all times and places, whether it is the Jim Crow Deep South, modern-day New York, or czarist St. Petersburg.

With that said, the inciting action of their decisions is a literary distinction.  Because it is a short story, where time is compressed, Fred Daniels' reasons for becoming a fugitive in the sewers and caves is much more immediate.  The story begins in medias res.  He is wrongly accused of a crime, murdering a white woman, that was punishable by quick and brutal justice.  Guilty or not, he is a perceived threat to the white men in the story.  The invisible man, however, is invited to speak about "social responsibility" by white leaders, only to be duped into fighting other black men and ogling a white ring girl in the battle royal.  He is the victim of manipulation and clever deception.  It's more of a hearts and minds campaign.

The differences between the characters are based on historical archetypes in slave literature.  Fred Daniels' initial plight is that of the field Negro, and his journey underground helps him discover his identity as a free man.  Ironically, he is shot for telling the truth, as he might have been shot as a fugitive.  The moral Wright seems to be making is that society is punishes the innocent and protects the guilty.  In Invisible Man, however, the narrator is a well-educated house Negro who realizes that he's been a repressed field Negro.   

"When I discover who I am, I'll be free," says the narrator of Invisible Man, newly awakened, suffering from amnesia. Ellison’s narrator later embraces yams, "They're my birthmark...I yam what I am!" In other words, a yam is a proud symbol and a negative stereotype of his Black Southern heritage.  He, wrongfully following the lead of Dr. Bledsoe, had shamefacedly associated soul food with "field niggerism," adopting instead to savor the more sophisticated delicacies of the house Negro (who caters for the white slave master) in order to escape the stereotype.

Since he has awakened, however, the narrator "no longer [feels] ashamed by things I had always loved"; he is no longer ashamed of his birthright, a poor black man from the Deep South.  After he takes a bite of the second yam, anticipating another sweet relic of home, it is frostbitten, leaving a bad taste in his mouth.  Although he naively loved the yams of home, the yams handed to him in his newfound freedom of the North are at first sweet, and then foul. He is in the process of reinventing himself, casting off both the house and field Negro stereotypes. His identity then is not defined by being inauthentic to any ascribed stereotype, but by relishing what he wants to relish, regardless of stereotype. 

Both Fred Daniels and invisible man champion underground freedom because it gives them time to write (Fred steals the typewriter and writes his name).  Both characters reject the materialism of the mainstream society (Fred papers his cave with worthless money).  I think both author's  responses to the American anti-black racial problem is not so much a social solution, but an existential one.  To live underground, or to be invisible, in a sense, is to be a conscious individual who cannot be predicted or manipulated, one who must reshape his racial identity from one obedient to authoritarian structures to one free to reject all of them.  Ellison’s narrator finally achieves freedom by refusing to take refuge in a false public image of himself.  Fred Daniels might have done the same, had he had the chance.