A curve \(f:I\to\mathbb R^n.\) is called closed if the interval \(I\) is closed, i.e. \(I=[a,b]\) and if \(f(a)=f(b)\).
A closed curve \(f:[a,b]\to\mathbb R^n\), whose restriction to the open real interval \((a,b)\), i.e. the function \({f|}_{(a,b)} : I \to \mathbb R^n\), is a simple curve, is called a simple closed curve.
The closed curve to the left is a simple closed curve, while the curve right to that curve is a closed curve, but not a simple closed curve.
(Image Source: bookofproofs)
(Image Source: bookofproofs)