Christians are not perfect and never will be, apart from the fullness of the Kingdom of God in glory. Yet, it is common that non-Christians accuse Christians of thinking that they (the Christians) are better than everyone else. And Christians are too often infected with the pride of thinking the same thing -- sometimes blatantly, sometimes subtly.
There is a sense in which this idea of moral superiority is true, and a sense in which it isn't. Christians do in fact grow and mature in morality, so they do enjoy a kind of moral upgrade. Moral improvement is one of the benefits of Christianity.
At the same time it is a supreme folly and sin for Christians to think that they have been saved because they are morally superior to anyone else. Too often the reverse is actually true. Many heathens are in fact morally superior to many Christians. The point is not that Christians are morally superior to heathens in general because some are and some aren't. Rather, the point is that Christians grow in moral purity. A Christian should always be more moral, more honest, more righteous than he or she used to be. We are not to compare ourselves with others (2 Corinthians 10:12) -- Christian or heathen, but with Jesus Christ. Jesus Christ alone is our model. (Comparison to others is not the same as imitation. See the chapter on “Imitation,” p. 61.)
While moral improvement is a result of salvation, it is not in any sense a cause or foundation of salvation. Rather, salvation is in Christ alone. Augustus Toplady was correct when he wrote:
Naked, come to Thee for dress; Helpless look to Thee for grace;
Foul, I to the fountain fly; Wash me, Savior, or I die.
At the same time, there is no salvation apart from moral improvement. Christians are "called to be saints" (v. 2, Romans 1:7) who are "sanctified in Christ Jesus" (v. 2). Christians grow in grace, grow in obedience, grow in faithfulness, grow in righteousness, grow in moral improvement. Christians are not necessarily better than anyone else, but they are necessarily better than they used to be.
And Christians cannot claim any personal credit (or glory) for their moral improvement. We are what we are, not because we have worked hard to become good Christians, but solely because Jesus Christ died on the cross and dispatched His Holy Spirit to us while we were still awash in sin and disobedience. The Holy Spirit grabbed us by the scruff of the neck and hauled us aboard the life raft known as Jesus Christ. We had nothing to do with it until we found ourselves in Christ. Good thing, too! Because we, like Israel before us, are completely unable in our own strength and ability to be what God has called all people to be. Nor are we now -- yet -- what God has called us to be. But Christ has satisfied God's demands, deflected God's wrath, and provided a way for us to grow in godliness.
Paul proclaims, it is the "Lord Jesus Christ, who will sustain you to the end" (1 Corinthians 1:7-8). Here is a clear expression of the doctrine of assurance. Our salvation is assured, not because of anything that we can do, but solely because of what Christ has done. Christ's grip on us is much stronger than our grip on Him. Our assurance of faith rests in His grip on us, not in our grip on Him.