Greg Gilbert speaks of confusing sin with sins in his great little book What is the Gospel?.
Gilbert writes:

There is a huge difference between understanding yourself to be guilty of sins, and knowing yourself to be guilty of sin. Most people have no problem at all admitting that they’ve committed sins (plural), at least so long as they can think about those sins as isolated little mistakes in an otherwise pretty good life— a parking ticket here or there on an otherwise clean record.
Sins don’t shock us much. We know they are there, we see them in ourselves and others every day, and we’ve gotten pretty used to them. What is shocking to us is when God shows us the sin that runs to the very depths of our hearts, the deep-running deposits of filth and corruption that we never knew existed in us and that we ourselves could never expunge. That’s how the Bible talks about the depth and darkness of our sin—it is in us and of us, not just on us.
On the second floor of the National Museum of Natural History in Washington, there is what is said to be the largest flawless quartz sphere in the entire world. The sphere is a little bigger than a basketball, and there is a not a single visible scratch, pockmark, or discoloration on the entire thing. It is perfect. People often think human nature is like that quartz sphere. Yes, every now and then we may smear it up with dirt and mud, but underneath the grime it remains as pristine as ever, and all we really need to do is wipe it clean in order to restore its brilliance.
The Bible’s picture of human nature, though, is not so pretty. According to Scripture, the sphere of human nature is not pristine at all, and the mud is not just smeared on the outside. On the contrary, we are shot through with sin. The cracks, mud, filth, and corruption go all the way to the center. We are, as Paul said, “by nature children of wrath” (Eph. 2:3). We are included in Adam’s guilt and corruption (Romans 5). Jesus taught this, too: “Out of the heart come evil thoughts, murder, adultery, sexual immorality, theft, false witness, slander” (Matt. 15:19). The sinful words you speak and sinful actions you do are not just isolated incidents. They rise out of the evil of your own heart.
Every part of our human existence is corrupted by sin and under its power. Our understanding, our personality, our feelings and emotions, and even our will are all enslaved to sin. So Paul says in Romans 8:7, “The mind that is set on the flesh is hostile to God, for it does not submit to God’s law; indeed, it cannot.” What a shocking and frightening statement! So thorough is sin’s rule over us—our minds, understanding, and will—that we see God’s glory and goodness, and we inevitably turn away from it in disgust.
It’s not enough to say that Jesus came to save us from sins, if what we mean by that is that he came to save us from our isolated mistakes. It’s only when we realize that our very nature is sinful—that we are indeed “dead in our trespasses and sins,” as Paul says (Eph. 2:1, 5)—that we see just how good the news is that there is a way to be saved.
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Greg Gilbert, What Is the Gospel? (Crossway Books), 54-55. Kindle Edition.