Also, how did the debate shape American institutions?
Both men were trying to develop an educational strategy to combat racism and economic inequality.
Du Bois formalized the idea of the "Talented Tenth". It was to educate 10% of black men in a classical education. These men would then become active in the struggle for equality. This approach was more confrontational because he wanted these men to be educated in the best institutions, institutions that were often times exclusively white, and the challenge the economic and political status quo. His book "Souls of Black Folk" lays out his ideas and examines the failings of reconstruction.
Washington emphasized technical education. That way black men would develop skills and then gain some economic power. He also emphasized making allies with like minded whites. His story and philosophy is laid out in his autobiography "Up from Slavery". His speech that was later called the "Atlanta Compromise" solidified his philosophy, in essence giving political power to southern whites in exchange for education for black men.
In general the beliefs and philosophy of Du Bois were adopted by black civil rights organizations of the 20th century. The NAACP, MLK movement, Malcom X, and even the Black Panther party, were all influenced by Du Bois. The hallmarks of the civil rights movement; right to vote, desegregation, access to political and economic power, we all things Du Bois wrote about 50 years before the Civil Rights Movement of the 50's and 60's.