The sin nature is an aspect of the doctrine of "original sin," but not the entire story.
Louis Berkhof's highly regarded Systematic Theology introduces the topic of Original Sin by calling it simply "the sinful state and condition in which men are born."
The Westminster Larger Catechism, answer 25, identifies three key components of original sin:
- The guilt of Adam's sin
- The lack of inherent righteousness
- The corruption of man's nature
The first of these isn't normally part of what we consider to be the "sin nature." Berkhof calls this original guilt, and distinguishes it from original pollution. The former expresses the state of mankind in relation to the law – guilty:
The guilt of Adam's sin, committed by him as the federal head of the human race, is imputed to all his descendants.
The latter, original pollution, includes the concepts of total depravity and total inability, which typically come to mind when considering the "sin nature." These refer to the inherent corruption of of man's nature, a lack of anything spiritually good in man, an inability to act in a way that meets the requirements of God's law, and an inability to come to love God on one's own.
So the simple answer to your question is that Original Sin is a broader term that includes the idea of the "sin nature," but it also includes the concept of "original guilt." Thus they are not synonyms.
As an aside, sometimes the word "original" can be confusing and lead to incorrect assumptions that it exclusively refers to the sin of Adam and Eve. Berkhof explains:
This sin is called "original sin," (1) because it is derived from the original root of the human race; (2) because it is present in the life of every individual from the time of his birth, and therefore cannot be regarded as the result of imitation; and (3) because it is the inward root of all the actual sins that defile the life of man. (2.2.4)
Thus the word "original" here should not be exclusively understood as referring to the first sin in history, but also to the "original" state of post-fall individuals before they have committed any actual sins.