Please explain, "Asymmetric is stronger than simply not symmetric".

If a relation is symmetric then there is a two way arrow, e.g. if someone is a blood relative of me then I HAVE to be a blood relative of them.

If a relation is not symmetric then there can be one arrow or two, e.g. if I like someone then they may or may not like me, either case could be true.

If a relation is Asymmetric then having one arrow means that there definitely cannot be two, e.g. if someone is my parent then I definitely CANNOT be their parent.

So basically the difference between non-symmetric and asymmetric is that in one we might have two arrows some of the time, but in the other we can NEVER have a second arrow once we have the first.

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.

In asymmetric and antisymmetric relationships no two distinct elements can be joined by arrows in the two directions. Moreover in asymmetric relationships there can be no loops.

Partial orders are antisymmetric. Strict orders are asymmetric.

Both conditions are generally different from the negation of symmetry; the latter means that there is at least one pair of distinct elements with an arrow in one direction but not the other.