For they being ignorant of God's righteousness, and seeking to establish their own righteousness, have not submitted to the righteousness of God. For Christ is the end of the law for righteousness to everyone who believes.
Rom 10:3-4 NKJV (emphasis mine)
How is it that man seeks to establish his own righteousness (self-righteousness)? By the Law. By his own works. By his own self effort. In seeking to establish his own righteousness by the things he does, he lives a life that is ignorant of God's righteousness! In seeking to establish his own righteousness, he has not submitted to God's righteousness. Is there not a big difference between man's righteousness and God's righteousness?
We cannot live up to God's righteousness by what we do. It's impossible. As James said (and as I pointed out in Big Difference 2), if you keep the whole blessed thing and stumble in just one point, you wind up being guilty of the whole thing because you are a transgressor and transgressors to not inherit the kingdom of heaven. You have fallen short of Jesus' words to "be perfect, as your Father in heaven is perfect." (Matt 5:48). You can even do all the things that Jesus commanded... and if you stumble in just one point, you have fallen short and you are guilty.
The good news is "the gospel!" In Romans 1:17, Paul is speaking of the gospel (see vs. 16) and he says, "For in it (the gospel) the righteousness of God is revealed..."
The gospel reveals God's righteousness, as opposed to man's righteousness that he tries to establish by what he does. The gospel replaces man's attempts at righteousness with the gift of righteousness.
*For an explanation of this Big Difference series, see the original post.*