As I understand it, it is well known that CRT can be used to show that $Z/p\#Z$ as a ring is isomorphic to the product of the finite fields $Z/qZ$ where $q$ ranges over the primes up to $p$.
For example, this principle is called in out the Wikipedia article on the Chinese Remainder Theorem as a restatement of the CRT in algebraic language.
I believe that I am clear on what CRT says and how it is used in the standard use. I am a bit hazy when it comes to restating the theorem in algebraic language and the advantage of doing this. It might be that an example of applying the CRT algebraically may do the trick to clarify what the algebraic version means.
What insights does the algebraic restatement offer that would not otherwise be available in the traditional statement of CRT?
What could we conclude if such an isomorphism did not exist? In other words, what do we gain from the restatement?
My goal here is to try to think more algebraically and to better leverage the insights from group theory and ring theory.
One answer is that the "algebraic" statement allows for a generalization to arbitrary rings (rather than just $\mathbb Z$), see further down in the Wikipedia article. This is useful is various contexts. The two most direct such generalizations are on the one hand principal ideal domains, where the CRT helps classify finitely generated modules, and on the other hand rings of integers in number fields, where it helps show for example that the ideal norm is multiplicative.
Even if we restrict ourselves to $\mathbb Z$ though, there are numerous applications. It basically reduces any question about finite abelian groups to cyclic ones, which are easy. The first thing that comes to mind is the fact that finite abelian groups are Pontrjagin self-dual, i.e. isomorphic to their character groups.