"Let not conscience make you linger,
Not of fitness fondly dream;
All the fitness He requireth
Is to feel your need of Him."
If we are to go to God with a right spirit, we must always understand our need of Him. The problem is that human beings constantly try to make themselves look better than they really are. Yes, it's true that we're created in God's image, and human beings do accomplish great things. But, it's also true that we're marred. And, from a spiritual perspective, we do no good apart from God. Even from a purely physical perspective, we have to admit that since a God exists, we are completely dependent on Him. Therefore, "In him we live and move and have our being" (Acts 17:28a). There is nothing that we can do apart from God's common and/or saving grace working in our lives.
So, what does this have to do with this final stanza of "Come Ye Sinners"? The first two lines share the sinful desire of human beings. When the author says "Let not conscience make you linger," the word has a different connotation than what you may think in today's vernacular. Conscience means: " the sense or consciousness of the moral goodness or blameworthiness of one's own conduct, intentions, or character together with a feeling of obligation to do right or be good." Therefore, the first two lines are almost synonymous statements. Basically put, stop thinking that you have or ever will attain to God's standard.
Think about it. If you depend on yourself for your own salvation, who are you glorifying? If you sit at home and worry about whether you've done enough to earn God's approval, who are you really trying to please and honor? God or you? This is suggested in the last two lines. God requires something, but it's not what you can bring to the table. It's what He can bring to the table.
You have needs. He has fulfillment. You have brokenness. He has wholeness. You have guilt. He has forgiveness.
All that God requires is that you feel, deep in your being, your need of Him. It's at that moment that a sinner also knows the deep mystery of being one of God's children - a saint.