One can only speculate as to why Washington and DuBois had different opinions about segregation and how to fight it; however one obvious difference between the two is that Washington had lived in the South and DuBois in the North following the Great Northern Migration at a time when Black Nationalism and consciousness were at the forefront of the movement for Black equality.
Booker T. Washington was one of the last of the slave-born generation to rise to prominence. He was supported by a number of powerful white politicians, which was probably a factor in his argument that Blacks should acquiesce to segregation and "separate but equal." They would thereby prove themselves worthy. It is entirely possible that Washington considered this the most pragmatic approach, as he did not believe his goals could be achieved without White support. He believed that because Blacks were outnumbered by Whites, the direct approach advocated by DuBois would backfire. The book Ragtime, contains a fictitious but telling illustration of Washington's position: When a young Black man kills a White and holes up in a library, Washington is sent in to get him to surrender. Washington tells the young man, "I promise you that your execution will be merciful."
W.E.B. duBois was born in Massachusetts, and had not shared Washington's experiences. His ideas were that one should inform the people and enforce constitutional rights through the courts. He was aided, no doubt, by the large concentration of Black voters in Northern areas which gave a strong voice to his program. Rather than argue that Blacks should acquiesce and seek cooperation from and with Whites, he argued vociferously against racism, and said that a small elite of African Americans whom he called the "talented tenth" would lead the bring about change.
Thus the primary reason for the difference beween the two is the circumstances of their birth and the areas in which they worked.