Q: What's the difference between the righteousness by faith and the righteousness by works?
A: One works and the other doesn't (pun intended).
Romans 9:30 What shall we say, then? That Gentiles who did not pursue righteousness have attained it, that is, a righteousness that is by faith; 31 but that Israel who pursued a law that would lead to righteousness did not succeed in reaching that law. 32 Why? Because they did not pursue it by faith, but as if it were based on works. They have stumbled over the stumbling stone,
Romans 10:2 For I bear them witness that they have a zeal for God, but not according to knowledge. 3 For, being ignorant of the righteousness of God, and seeking to establish their own, they did not submit to God's righteousness. 4 For Christ is the end of the law for righteousness to everyone who believes.