Question: What's different about applied mathematics that you don't enjoy?
Notice the difference between that and asking why they don't like applied mathematics. One demands an accurate description of applied mathematics, the other can be answered in terms of one's aspirations or perception of value, and that distinction can be explained if it's unclear or not acknowledged.
Deep theorems often come from intuitions. Intuitions have to come from somewhere. Sometimes you get them from working elementary examples' features into your brain, but this can be near-sighted. Instead, they often come from an appeal to a natural connection between two subjects, and the overdetermining of properties induced by the multiple sets of semantics - every group is a permutation group, for example, is a formalization of the intuition about groups being transformers as well as algebras. The Sylow theorems and several properties of interest are capturing periodicity and self-similarity decompositions of group structure arising from viewing them as or expecting them to be the collections of symmetries of an object. Applied mathematics functions always in a context, and cross-pollination of semantics is an inherent part of such naturalistic environments. As such, it's an important source of intuition.
Projective geometry, for instance, arose from applied mathematics and the intuition it brought, and its importance in the structure of abstract objects (algebraic geometry, projective representations, manifolds, homology, hyperbolic geometry, continued fractions, the modular group) is worth the thinking it takes to justify without any trace of solipsism.
Applied mathematics can lead to these new fields because it poses problems that wouldn't be considered problems or be thought up by someone working only off their existing knowledge. It can also pose problems that are unmanageably difficult, which in mathematics suggests there are underlying paradigms left to be uncovered.
For a modern example, crack analysis can now be done using fractal measures and other abstract ways of reasoning, except when it comes to intersections of propagating cracks, these theories tend to break down abruptly and the finite-element method must be used to proceed with the description of the cracks approximately.
It's also one of the best ways to motivate a special case, informing the specific calculations that will allow one to become adept at a field. Most (classes of) differential equations are posed as tasks for their representation of a physically observed behavior, and in that way arbitrary calculations obtain a life of their own through application. So, for a pure mathematician - and by that I mean one which isn't strictly concerned with real-world tasks, applied contexts are rich interfaces to mathematics. Knowing any field in such detail helps gauge your depth of knowledge in other fields, and an application will evade giving a sense of security until that point is reached and the motivating problems have been solved.
All mathematics is descriptive of something, so how is non-applied mathematics to be intrinsically distinct from applied mathematics? It might be they disagree with having a non-mathematical goal, or have the impression that someone primarily does applied mathematics when mathematical goals aren't inherently valuable to them. It might be because educators often treat students as if mathematics is opaque to them whenever it isn't shrouded in a real-world scenario. These are cultural reactions. Perhaps their problem is actually with the experience they expect of applied mathematics, or the type of output they expect to be giving. The diagnostic should come before the prescription. Much of what people are trying to avoid by professing these alignments can be a problem in pure mathematics, in a generalized form, and if you have the wrong reasons to avoid something you narrow your tools for gaining experience and proficiency in a field in which you're mostly left to your own devices.