Booker T. Washington uses the metaphor of the fingers and the hand to alleviate the pressures felt by both whites and blacks. Whites did not want to feel forced into interaction while a lot of blacks would have probably felt resentment towards having to interact with whites. In the passage preceding this declaration, he states, "we shall stand by you with a devotion that no foreigner can approach interlacing our industrial, commercial, civil and religious life with yours in a way that shall make the interests of both races one."
He directly communicates and uses the word civil in an effort to show that he is for equality in facets of society where it is most prevalent. Social appears to mean all aspects in which interaction among African Americans is recreational, like going to a play or movie or eating at a restaurant. Furthermore, I think Washington gives leeway in case things do change in that if it so happens that what is social becomes of civil interest, that too, would become apart of the hand of mutual progress.
In the case of W.E.B. DuBois, he pairs the word civil with political equality in several instances which gives the reader the impression that they should go hand in hand. When Washington uses social, DuBois infers that he means civil and political rights that have a distinct implication of law and government which I believe is farthest away from Washington's argument. I also believe he misconstrues Washington's intent thereby making it easier for him to be criticized.
His choice of using the word civil or civic ingrains in the reader that African Americans are citizens, and thereby should be able to participate in the rights allocated to every citizen of the United States. This is a good point but I don't think it is comparable to Booker T. Washington's "social".