Yes, the sin of the world (singular) is different from sins (plural). All the sins of mankind are rooted in "the sin". You can look at it like this, "the sin" is the root of all sins. "Sins" are the fruit and "the sin" is the root. John, speaking of Jesus, said
Matthew 3:10 Even now the axe is laid to the root of the trees. Every tree, therefore, that does not bear good fruit is cut down and thrown into the fire.
"The sin of the world" or simply "the sin" is what Jesus came to deal with. Romans 5 and 6 speak of "the sin" that is dealt with in being crucified with Christ. The Youngs Literal Translation is the only English translation that correctly adds the definite article before sin, distinguishing "the sin" and "sin".
Romans 6:1-14 What, then, shall we say? shall we continue in the sin that the grace may abound?
2 let it not be! we who died to the sin -- how shall we still live in it?
3 are ye ignorant that we, as many as were baptized to Christ Jesus, to his death were baptized?
4 we were buried together, then, with him through the baptism to the death, that even as Christ was raised up out of the dead through the glory of the Father, so also we in newness of life might walk.
5 For, if we have become planted together to the likeness of his death, [so] also we shall be of the rising again;
6 this knowing, that our old man was crucified with [him], that the body of the sin may be made useless, for our no longer serving the sin;
7 for he who hath died hath been set free from the sin.
8 And if we died with Christ, we believe that we also shall live with him,
9 knowing that Christ, having been raised up out of the dead, doth no more die, death over him hath no more lordship;
10 for in that he died, to the sin he died once, and in that he liveth, he liveth to God;
11 so also ye, reckon yourselves to be dead indeed to the sin, and living to God in Jesus Christ our Lord.
12 Let not then the sin reign in your mortal body, to obey it in its desires;
13 neither present ye your members instruments of unrighteousness to the sin, but present yourselves to God as living out of the dead, and your members instruments of righteousness to God;
14 for sin over you shall not have lordship, for ye are not under law, but under grace.
To understand what "the sin" is, you first must understand a few things about the government of God and the original sin. First off, when Jesus came preaching the gospel he said,
Matthew 4:17 Repent, for the kingdom of God is at hand.
Strange thing to say. Seems that he should have said, "Believe that I'm going to die and rise again on the third day to be forgiven and you'll go to heaven when you die." But instead, he preaches repentance. Not for doing bad things necessarily, but because there is a kingdom that is present. Why should one repent because the kingdom of God is at hand? A kingdom is a form of government. A monarchy. Where one person rules. That person, a King, makes up the rules and culture of their kingdom. Two people cannot sit as king in a kingdom. Jesus said,
Matthew 12:25 But Jesus knew their thoughts, and said to them: “Every kingdom divided against itself is brought to desolation, and every city or house divided against itself will not stand.
When Jesus says to repent for the kingdom of heaven is at hand, he is saying to repent of being the authority of our own lives. This is the original sin.
You see, the kingdom of God was prepared before the foundations of the earth were laid.
Matthew 25:34 Then the King will say to those on his right, ‘Come, you who are blessed by my Father, inherit the kingdom prepared for you from the foundation of the world.
So before Genesis 1:1 God created a government, a kingdom. In that kingdom, God created seraphim, cherubim, angels, etc. One of those was Lucifer. Isaiah prophecies about the fall of Lucifer and how the kingdom of darkness originates in Isaiah 14:12-15. He is prophesying to the king of Babylon and as he is prophesying he speaks to the spiritual influence of the king of Babylon, Lucifer, and declares how he fell.
Isaiah 14:12-15 ESV “How you are fallen from heaven,
O Day Star, son of Dawn!
How you are cut down to the ground,
you who laid the nations low!
13You said in your heart,
‘I will ascend to heaven;
above the stars of God
I will set my throne on high;
I will sit on the mount of assembly
in the far reaches of the north;
14 I will ascend above the heights of the clouds;
I will make myself like the Most High.’
15 But you are brought down to Sheol,
to the far reaches of the pit.
Notice Lucifers sin. Five times he says in his heart "I will...". Why does this get him cast out? Because he is in a kingdom. He commits high treason by no longer desiring to do the will of the King, but instead, he is committed only to his own will. He sets himself up as ruler/judge/king/lord of his own self, thus committing treason against God's will. This is "the sin" that leads to all other sins. This is "the sin" that leads to death. (Romans 6:23)
This is the same sin that Adam and Eve were tempted with and that every human being was born into since then.
When the serpent tempted Eve he tempted her with becoming like God in the sense of being her own judge of what was good and what was evil. Being her own ruler/king/judge/lord. People are often mistaken in that they believe Adam and Eve didn't know right or wrong before the fall. But this is clearly not the case for Eve knows that it is wrong for her to eat of the tree of the knowledge of good and evil. God has already told them so.
Genesis 3:2-3 ESV And the woman said to the serpent, “We may eat of the fruit of the trees in the garden, 3 but God said, ‘You shall not eat of the fruit of the tree that is in the midst of the garden, neither shall you touch it, lest you die.’”
Their sin was that they decided they would be the judge of what was good and what was evil for themselves. They would become like God in this sense. This is the sin that has lead mankind into all the evil, hurt, injustice, perversion, and sin throughout human history. This "natural instinct" to be the deciders of what is good and evil, which is in all of us, is what Jesus came to take away as the Lamb of God. If he deals with the root, naturally the fruit will be dealt with.