What is the relation between pure mathematics, applied mathematics and Cellular Automata?
The definitions given in Wikipedia are trying to convey a very abstract concept by means of examples, which, to my taste, isn't satisfactory.

Thus, before answering this question we'd need to ask ourselves why are there two in the first place.  And if there is a real need for two, then:


  How is mathematics (both pure and applied) possible?
  -- Immanuel Kant


History

The answer to the first question appears to be very involved.  In European tradition, the question goes back to Plato's distinction between the ideal and the concrete.  These roughly translate into analytic and synthetic in modern philosophical language.  Plato ultimately believed that mathematics is discovered rather than constructed, but this still puts mathematics in the realm of ideal due to Plato believing that only the ideal actually exists, and that the concrete is but an imperfect reflection of the ideal.  This was in stark contrast to the existing tradition of viewing mathematics as a practical skill of measuring and counting.

Since that time on, multiple attempts were made to either unify both, or to define one in terms of the other, or to completely dismiss one of the two.  Ultimately, in the beginning of the 20'th century the argument boiled down to the ability of defining the classical mathematics.  This split the philosophy of mathematics into logicism (with Frege as its forerunner), formalism (with Hilbert as its advocate) and psychologism (intutionism), the idea advanced by Brouwer.  Somewhat under-represented here are the universalist views held by Klein (no good online quotes, sorry).  The later is the view that synthetic and analytic truth are the same, and that when a seeming contradiction arises, it is probably because we either don't have a proper observation technique, or that we don't have a proper mathematical explanation for the observation.

The Problem

It all boils down to the question of what is truth, and are there different kinds of truths?  I will first give an example of the problem and then discuss it.  

Stating that

$$
    \textit{bachelors are males}
$$

Imposes a restriction that the interpretation of "bachelors" must be a subset of the interpretation of "males", but this restriction doesn't result from logical construction of the sentence, rather it is our experience (and the reasoning non-included in this sentence) that all bachelors we've seen so far, and are likely to meet in the future will be male.  This is an example of synthetic truth.  While in this example it may look like it could be possible to resolve the problem by adding more logical predicates, the problem is in fact unresolved as of now.  More generally, this can be seen as the disparity between material implication and "real" implication.

Using material implication, typically denoted as $\implies$ we can obtain:

$$
\textit{water is wet} \implies \textit{sky is blue}
$$

And while the sentence is true in a pure logical sense.  From the perspective of non-logical, or extra-logical reasoning it is false, since the perceived properties of water have no immediate impact on the perceived color of the sky.

Lewis was, perhaps, the first to notice and try to address this general problem by creating more restricting logical systems.  Another way to approach this problem, and the stance taken by many contemporary philosophers is laid out in Two Dogmas by Quine.  To quote the later:


  Physical objects, small and large, are not the only posits. Forces are another example; and indeed we are told nowadays that the boundary between energy and matter is obsolete. Moreover, the abstract entities which are the substance of mathematics -- ultimately classes and classes of classes and so on up -- are another posit in the same spirit. Epistemologically these are myths on the same footing with physical objects and gods, neither better nor worse except for differences in the degree to which they expedite our dealings with sense experiences.


Put simpler: "pure" mathematics is just a convenience, a make-believe thing that doesn't have any factual content and has no bearings on what truth is.  Surprising as it may sound, we still use the terminology in its pre-Quinean sense.

Finally

When forced to define abstract and applied mathematics, I'd say that:


Applied mathematics is pure mathematics restricted to the synthetic truth.  I.e. the kind of mathematics that can be verified by experimentation.
Pure mathematics is the art of discovery of patterns.  The parallel between art and mathematics is not new, and, in fact, Hardy, the iconic pure mathematician holds that:


  I am interested in mathematics only as a creative art. 



Your question

So, having dealt with preliminaries :) to answer your question!


Pure mathematics can find more than one interesting aspect of cellular automata to study.  There are so-called "free objects" in mathematics - those possess only one interesting property which defines them.  CA isn't one of those.  It is a whole area of research, so any example of its treatment in pure mathematics will not be exhaustive.  One such (non-exhaustive!) example is to represent this particular CA as a function in domains being CA's input and output, and study the function's behavior in the usual way (answering questions about totality, bijection etc.)
Pascal's triangle can be a subject of applied mathematics as you can easily build a physical model for it and convince yourself of the calculation that you do by means of counting the elements of the model.  Since, at least from my perspective, applied mathematics is the restricted version of pure mathematics, this means that regardless of your ability to build a physical model, this is still valid pure mathematics.


PS

The views expressed above are certainly biased and don't represent the absolute majority of the people involved in this field.  You would benefit a lot from reading literature dealing with subject matter.