Original sin is the doctrine that explains why we are born sinners. I did not deserve to be born a sinner but since the guilt and condemnation of Adam's sin was charged against all humanity in Adam, I was born guilty and deserved to be born a sinner due to Adam's sin. In the same way, I do not deserved to be counted righteous and sinless when believing in Christ, yet his righteousness is imputed to be and I obtain life through the gospel. The idea of being judged by the actions of another Adam, or Christ, humanities two federal heads is the basis of the gospel.
The sinful nature is the result of Original sin. It means that as soon as a child is able to make a moral decisions it will be an evil one. As soon as a human knows right from wrong a human will do what is wrong. In other words as soon as a baby knows what it is to love God, the baby will not love God with it's whole heart. Yet there is more.
The sinful nature is not just a result of original sin but is a result of being found obnoxious and guilty under God's moral law. Therefore Christ must find a way to remove the law's ability to condemn us to restore the nature from its sinfulness. As long as the law has a right to judge, we will be cast into a fully corrupt nature. Only by removing the law's right to judge us can our nature be free from original sin and its guilt. This again is by Christ fulfilling the demands of the law and providing his perfect righteousness to us, so that under the law we are considered perfect. As the law no longer casts us into a sinful nature we are free from sin. Therefore the only way a human can do good is to first have the law removed through the death of Christ and the imputation of his righteousness to sinners. In fact the moral commands of the law was added that our sinful nature would become more sinful and thus be made known to our minds, revealing our sinful nature.
Non-Protestant churches accept a form of this doctrine called 'justification' but reduce its meaning to nothing by saying it is a life time process that you can ruin any time you wish. Protestants teach that this justification happens in a moment when a sinner believes in the gospel. In fact this is the greatest point of division between any kind of theology about God that can ever be made and the most important point to be decided on by a student of scripture.
Here is a good summary of what I am arguing and the distinction of the concepts:
For if the many died by the trespass of the one man, how much more did God’s grace and the gift that came by the grace of the one man, Jesus Christ, overflow to the many! 16 Nor can the gift of God be compared with the result of one man’s sin: The judgment followed one sin and brought condemnation, but the gift followed many trespasses and brought justification. 17 For if, by the trespass of the one man, death reigned through that one man, how much more will those who receive God’s abundant provision of grace and of the gift of righteousness reign in life through the one man, Jesus Christ!
18 Consequently, just as one trespass resulted in condemnation for all people, so also one righteous act resulted in justification and life for all people. 19 For just as through the disobedience of the one man the many were made sinners, so also through the obedience of the one man the many will be made righteous.
20 The law was brought in so that the trespass might increase. But where sin increased, grace increased all the more, 21 so that, just as sin reigned in death, so also grace might reign through righteousness to bring eternal life through Jesus Christ our Lord. (NIV, Romans, 5:15–21)