Why are we interested in permutahedra, associahedra, cyclohedra, ...?
Philosophical questions deserve philosophical answers, so I am afraid no amount of references and specific results will probably satisfy you. Let me try to explain it in a somewhat generic way.
Think about it this way - why care about sequences like $\{n!\}$, Fibonacci or Catalan numbers? The honest answer is "because they come up all the time". Now, once you know these sequences, you may want to understand the underlying structures (permutations, trees, Dyck paths, triangulations, etc.) You may then want to understand connections between structures (e.g. bijections), algebraic or geometric interpretations (e.g. group representations, volumes of polytopes), etc. Once you have developed some kind of structures you may want to understand the relations between different structures, whether your bijections are structure-preserving, etc. That's how you develop the theory starting with just numbers!
In general, basic objects in combinatorics tend to lack structure. Adding structures is always welcome as they present a deeper understanding of the underlying objects (and sometimes even just numbers). It's what allows to employ and further develop tools from other parts of Combinatorics and other fields. This is the setup in which one can understand results such as Kuperberg's proof of the number of ASMs or the Adiprasito-Huh-Katz theorem, but it doesn't have to be so spectacular. Sometimes even a weak structure can lead to unexpected connections and generalizations unforeseen otherwise.
In summary, "these polytopes are just further examples of polytopes" is a misunderstanding of the context in the same way as Fibonacci and Catalan numbers are not "just numbers". Viewed in context, permutahedra and associahedra exhibit structures of combinatorial objects invisible otherwise.
In my opinion there are two answers to this question.
The first is that these particular classes of polytopes have fascinating combinatorial properties and structure. Presumably you're aware of the work of Postnikov and others in this direction. In my view, and the view of many others, these properties make the polytopes worth studying in their own right.
The second is that these polytopes arise naturally in other contexts, e.g., as moment polytopes (images of certain manifolds/varieties under the moment map) and as weight polytopes (convex hulls of subsets of the weight lattice of certain Lie groups). Various geometric properties of the manifolds, and representation-theoretic properties of the group, can be reduced to combinatorial properties of the polytopes. For example, GKM theory tells you that (provided certain hypotheses are satisfied) you can define and calculate equivariant cohomology directly on the moment graph.
This of course is a very high-level answer. If you are wondering about whether certain specific combinatorial properties have geometric or representation-theoretic significance, then maybe you can state a more specific question.
There are remarkable combinatorial formulas for the face numbers and the volumes (of certain geometric realizations of) of these polytopes and a more general family ("generalized permutohedra" a.k.a. "polymatroids") to which they belong. These numbers include classical sequences like the Eulerian numbers, Catalan numbers, $(n+1)^{n-1}$, etc. This is a major combinatorial interest in these polytopes.
Regarding the question about why geometric realizations are interesting in particular, note that to meaningfully talk about volume (and its relatives like discrete volume, i.e., number of lattice points, etc.) requires a particular geometric realization.
(Warning: "generalized permutohedra" and "generalized associahedra" are two different families of polytopes, despite the very similar names. "Generalized permutohedra" are obtained by deforming the normal fan of the regular permutohedron, i.e., sliding facets in and out along a normal vector. "Generalized associahedra" have to do with analogs of the associahedron in other Lie types.)