Continuous chemical reaction network versus chemical reaction network - difference?
I suspect that the distinction is that a CRN considers discrete species concentrations whereas a CCRN considers continuous species concentrations: "The CCRN model is continuous, dealing with real-valued concentrations of species...". More precisely, whereas a CRN deals with concentrations $x(t), y(t) \in \mathbb{Q}$, a CCRN deals with concentrations $x(t), y(t) \in \mathbb{R}$.

On a microscopic level, it's easy to see why $x(t), y(t)$ should be elements of $\mathbb{Q}$: a microscopic concentration is simply a ratio of the number of one molecule to the total number of molecules in the system, and the numerator and denominator must thus be integral. 

For macroscopic systems, however, since the denominator---the total number of molecules in the system---is so large, it becomes a very good approximation to take $x(t), y(t) \in \mathbb{R}$. Moving to $\mathbb{R}$ is convenient, since then by physical considerations $x(t)$ and $y(t)$ are continuous, and we can, under mild assumptions, take derivatives and return to a rate-law formalism involving differential equations rather than difference equations.



Remark. I'm not familiar with this field, and I don't know how common the CRN/CCRN distinction is. Indeed, the Wikipedia article assumes differentiability of $x(t), y(t)$ and hence that $x(t), y(t) \in \mathbb{R}$.