First, let's note that the terms as used by Rosen are standard definitions, as we can see on Wikipedia (here and here), as well as other resource sites. There was some question about this in the comments, so I thought to clarify this first. Perhaps reading those articles will give an added perspective for the OP.
Now, I'm not going to offer a mnemonic -- I don't think it's a good practice. I almost always find there is some deeper meaning to mathematical structures, which when understood makes the relationships much clearer and makes a mnemonic unnecessary baggage. Usually I find that students reliant on mnemonic devices use them as a crutch, barely succeed in the current course of study, and fail to succeed at a later step.
That said, here are some comments looking at the Rosen text (speaking of Kenneth Rosen, Discrete Mathematics and its Applications, Seventh Edition) that may be clarifying.
In Section 9.1, the definition of antisymmetric appears in the main text, whereas asymmetric only appears within the exercises of that section. I would highly encourage you to work through each of Exercises 18-24 in that section, as the sequence walks you through clarifying exactly what the difference is. (The term asymmetric is only referenced in 4 other exercises throughout the rest of the text; it is largely only presented to walk through the sequence reflecting on the exact meaning of antisymmetric.)
Spoilers follow for those exercises.
For what it's worth, I don't think that the phrasing of the antisymmetric definition there is the best it could possibly be. When I present it in my class using that book, I rephrase it as the following (and this may be the only time I rephrase any definition in other than the exact way the book does):
Antisymmetric: A relation $R$ on a set $A$ such that whenever $(a, b)
\in R$, then $(b, a) \not \in R$ (except when $a = b$).
Note that while the Wikipedia article gives two equivalent ways of phrasing this, my version shows a third way that I think is a bit clearer. Note that this is now obviously saying the same thing as the definition for asymmetry, with the added clause "except when $a = b$" at the end.
So this highlights that antisymmetry and asymmetry are almost the same thing, through which Exercises 9.1 walk the student. Any asymmetric relation is necessarily antisymmetric; but the converse does not hold. Specifically, the definition of antisymmetry permits a relation element of the form $(a, a)$, whereas asymmetry forbids that. So an asymmetric relation is necessarily irreflexive.
The property of antisymmetry is fundamentally of more interest, because it is used in the definition of a partial order, that is, a poset. See the start of Section 9.6: "A relation $R$ on a set $S$ is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive." See also Wikipedia here.
The property of asymmetry is not as interesting in this sense, because it's not independent of the reflexive property; as noted above, any asymmetric relation is automatically irreflexive.
Perhaps working through Section 9.6 on your own, and seeing the major use-case for antisymmetry (as opposed to asymmetry), will also be a memorable exercise.