A strong antisymmetry is in particular weak because on the conditional statement that defines the weak antisymmetry, a strong relation will never satisfy the antecedent, thus making the conditional statement true. Therefore any strong antisymmetry is in particular weak.
It's useful to think of a weak antisymmetry as $\leq$, for whatever intuitive meaning of $\leq$ you're more comfortable with (for example the order relation on the real numbers). Similarly you can think of a strong antisymmetry as being $<$.
You can also observe that any strong antisymmetry can be extended to a weak antisymmetry by adding elements to the strong antisymmetry, but you can't extend a weak antisymmetry to a strong antisymmetry because a strong antisymmetry can't be reflexive - actually it doesn't even get to the point of a strong antisymmetry not being reflexive, it's stronger than that, it actually is antireflexive.
The fact that a strong antisymmetry is also weak, does not at all imply that is must have all the properties that a weak antisymmetry would have. I think the contrary is more intuitive, the broader concept is the one who will have all the properties of the smaller concept (this isn't true either, by the way).