Paul, embroiled in a situation where he was being judged for the way he conducted his affairs as God's apostle, gives some excellent advice. Passing judgment on someone based on our narrow perspective and subjectivity is an exercise in futility and vain, with nothing of spiritual value to be gained from it. That is why God does not want us doing it. Its prime motivator is to elevate (cf. verse 6) or justify the self.
Paul did not even pass judgment on himself! He certainly examined himself because he wrote to this same church, "Examine yourselves as to whether you are in the faith. Prove yourselves" (II Corinthians 13:5). Then why did he say he did not judge himself? Because we all are saved by grace through faith. We certainly are not saved by our own favorable judgment of our conduct. Though he could find nothing wrong with his conduct in this situation, he still would not step into Christ's area of authority as Judge. Even his blamelessness did not justify him.
Many things we judge in others are trivial and sometimes extremely "picky." Many situations do not involve sin at all but simply different ways of doing things. We tend to pounce on situations or characteristics that will hardly mean a thing a year from now—and certainly will matter nothing in a thousand years. There may be nothing wrong with pointing them out to someone concerned, but why focus on them to the point we pass judgment on the person?