Washington was born into slavery in Virginia and did not know the identity of his father, but did know that he was white. Nothing was handed to Washington, he had to struggle for every bit of success he ever achieved. His "struggle for an education" began when he decided to attend a trade school in Virginia. While he attended the Hampton Institute, Washington worked as a servant in exchange for his education. Perhaps this is where he obtained his philosophy for racial equality. Washington believed that in order to achieve social equality, African Americans must first achieve economic equality. He believed that in order to achieve equality African Americans must work together with whites. Washington obtained his education and was able to achieve success through hard work and cooperation, so it is easy to see why he thinks this is the key to racial equality.
Du Bois, on the opposite end of the spectrum, was born in Massachusetts. The town Du Bois grew up in was a small town that was fairly secluded from the segregation that was rampant in the south. After he graduated High School, Du Bois went south to Fisk University to attend college. It was there that he experienced segregation and racism in a way that he had never experienced it before. This is likely where Du Bois developed his extreme separatist attitude. He was highly opposed to Booker T. Washington's philosophy of accommodation and economic equality coming before social equality. He believed that African Americans had been laying down arms for too long, and it was time to turn to the extremes.
Du Bois and Washington had two completely different experiences growing up and going to school. Learning about both of their backgrounds allows you to read their works and understand their positions on racial equality much better. It's not that either man was in the wrong, nor was either one of them completely in the right. They were just two completely different men who led two completely different lives. These separate paths shaped their views about their race and gave them different perspectives as to how to achieve racial equality.