How is Darnay's philosophy different from his uncles' in book two chapter 9?
I think that the differences between Darnay and his uncle, the Marquis, are deeper than merely the circumstances of their lives. Darnay appears to be deeply influenced by the Enlightenment Philosophy of the 18th and 19th centuries, led by thinkers such as Locke, Rousseau, and Voltaire. Enlightenment philosophy held that all men are equal before their creator and that all men are responsible for contributing to society. Human reason was elevated above all else. We can see the reflection of these philosophies in Darnay's rejection of his family fortune which was based on birth rather than ability, and his desire to earn his own way in society. We can also see it in his concern for the peasantry under his uncle's control and in his determination to help his steward even though it puts him in danger.
His uncle, the Marquis, reflects the older, more medieval school of thought which maintains that certain rights and privileges come to a man because of his birth. He believed that his noble blood made him superior to those beneath his station and that this was a part of God's plan. In earlier medieval thought, these privileges came with distinct responsibilites as well, but by the Marquis' time the concept of noble obligation was lost and only the privileges remained. Thus each represented a distinct philosophy which would have been known to Dicken's contemporary readers.
Charles Darnay wants to renounce his name and birthright. He is ashamed of what his family name has stood for-fear and slavery. Darnay goes on to point out that the family destroyed anything that came between them and pleasure. The Marquis finds this outrageous and believes he should accept his natural destiny. He relishes the fact that the common man are afraid of him.