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.
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.)