The biggest difference between these two characters is their position in regards to white society and their own black identity. It is clear that George is a character who has become successful through making himself conform to white standards as much as possible. He works hard and thinks that white dominance in society means that you have to play their game to get ahead. He also has no interest in promoting a black identity as is shown by his horror when Beneatha cuts her hair so drastically. In addition, he feels it is impossible to enact change and the only thing blacks can do is to work hard and get a degree to get ahead with the world. This is shown by his impatience with Beneatha's thoughts of identity and change:
It's simple. You read books--to learn facts--to get grades--to pass the course--to get a degree. That's all--it has nothing to do with thoughts.
His is a very pragmatic, matter-of-fact view of the world.
Joseph, by contrast, is a character that is all about creating and promoting an African identity, as his shown through his constant references to Africa, his giving Beneatha an African name and the way that he encourages her to wear her hair in the African manner rather than the western manner. He encourages Beneatha to explore her African roots and presents an opposing view to identity from the view of George.
The two characters are used in the play to express two different responses to white dominance in society: accommodation and modification, as expressed by George, or creating a new and separate identity based on rediscovering an African past, as expressed by Joseph.