In the closing verses of the psalm, David turns from examining the heavens and the Scriptures to look at his own heart. He realizes there are depths to his own soul he cannot plumb. "Who can understand his own errors?," he asks.
One thing I find interesting is that verses 12, 13, and 14 seem to describe a progression in sin from "hidden faults" to "presumptuous sins" to "great transgression." All sins are sins; all sins deserve God's wrath; and yet not all sins are of the same magnitude.
David wants God to cleanse him of hidden faults precisely because hidden sins -- sins we hide from ourselves, but also sins we think we can keep secret from others -- grow into much greater sins if left unchecked. If God will deliver David from his hidden (secret, private) sins, that will keep him from falling under the dominion of presumptuous sins. And if the Lord will keep David from these presumptuous sins, that will keep David from falling into even more disastrous "great transgressions."
Surely, we have seen this pattern in our own lives (and perhaps we have even seen it play out in the lives of others): a small sin goes undealt with, it grows into an even larger sin, and then becomes a great transgression. Sin, like cancer, can grow if not dealt with ruthlessly. When someone falls into great transgression, it is almost always the case that the big sin grew out of much smaller sins, perhaps even private sins, that were never discerned and put to death. Adultery often started with a lustful thought or a secret visit to an illicit website; sin was not dealt with in its smaller form so it grew to disastrous, life-destroying proportions. Broken relationships often started as small grudges, but hurt feelings grew into bitterness, and bitterness was allowed to grow into full-fledged hatred.
At all times, but especially during this Lenten season, let's ask God to deliver us from small and hidden sins so they are not allowed to grow into larger sins that would enslave and wreck us. Examine your conscience and do not tolerate sin's progression in your life. Stamp out sinful thoughts and desires before they have a chance to grow. Pull up weeds from the soil of your heart before their poisonous fruit is manifest. Follow John Owen's counsel: Kill sin before it kills you (Romans 8:13). We re all locked into a battle unto death with our great foe, the sin that dwells in our hearts. We better to fight to win -- and David's prayer at the end of Psalm 19 shows us how.