Did W.E.B. Du Bois and Booker T. Washington agree?

Both advocated for the advancement of black people but strongly disagreed on the method of achieving it. 
Washington, founder of the Tuskegee Institute, favored what he called "practical knowledge" over the liberal arts education that Du Bois had acquired. He eschewed civil rights and, during a speech at the Atlanta Exposition, famously claimed that blacks and whites could exist socially as "separate as the fingers, yet one as the hand in all things essential to mutual progress." The Atlanta address is sometimes also referred to as the Atlanta Compromise, for that was exactly Washington's intent: to placate white fears about integration, while also, ever so cleverly, asserting the necessity of black participation in economic life. He encouraged black people to cast "down your buckets where you are," which was a message of self-reliance. 
Tuskegee, like several other historically black colleges and universities (HBCUs), was initiated as a trade school, in keeping with Washington's preference for practical knowledge. This mindset was not completely intended to placate whites. Privately, Washington was not the deep thinker that Du Bois was. For example, he disliked reading fiction; he thought stories were a waste of time. 
Nevertheless, he had strong supporters in white leaders, particularly philanthropists who helped fund Tuskegee. He opened the school with merely $2,000 in funding from the state of Alabama. Twenty-five years later, the school had 1,500 students enrolled, training in thirty-seven industries. He was also invited to the White House, where he advised Presidents Theodore Roosevelt and William Howard Taft on race relations. 
Du Bois despised everything that Tuskegee represented, due to its subordination of independent and critical thinking in favor of manual skills. However, Washington's displacement from black leadership—in favor of Du Bois—had less to do with Du Bois's brilliance and more to do with timing.
After the First World War, thousands of black soldiers returned to a nation still intent on treating them as subhuman, despite their immense sacrifice. Knowing that life in Europe, particularly in France, was freer, many black Americans now questioned the validity of Washington's ideas, which seemed regressive and irrelevant. 
Du Bois's criticism of Washington began in 1901. In a review for Dial magazine, Du Bois encouraged a break between conservative and "radical" wings in the black community, with Washington representing the former and Du Bois, the latter. The intent of this was probably not only ideological, but self-promotional and intended to diminish Washington's credibility. Du Bois accused Washington of using his power to silence his critics and manipulate black media.
Du Bois praised the Atlanta Exposition address for its eloquence, but, in The Souls of Black Folk, derided the passivity encouraged by Washington. In the chapter, "On the Training of Black Men," he promoted his idea of "the Talented Tenth," a liberally-educated group of black people who would assume leadership positions in their respective communities. This leadership would be politically motivated. In 1905, Du Bois established the political and social action group, the Niagara Movement, which evolved into the NAACP.