Please explain, "Asymmetric is stronger than simply not symmetric". In some textbook I found a statement like, "Asymmetric is stronger than simply not symmetric".
But as I try to perceive this statement, both appear to be same to me.
For example, parentof is an asymmetric relation. If $A$ is a parentof $B$, $B$ can not be parentof $A$. We arrive at the same conclusion if we call this relation to be not symmetric.
(Liyang Yu. A Developer’s Guide to the Semantic Web.
2nd. Springer, 2014. p 226, last sentence of second paragraph from bottom.)

It is just the standard negation of quantifiers: Asking that a relation is never true is stronger than the negation of the relation being always true.
A relation being symmetric means that all pairs can be inverted.
The negation of this is that some pairs cannot be inverted.
Asymmetric means that all pairs cannot be inverted.
