Alternative notations include $C(n, k), nC_k, ^nC_k, C^k_n, C^n_k, C{n,k}$ in all of which the C stands for combinations or choices. -from Binomial coefficient - Wikipedia

As you can see, the C notation varied from country to country.
So using it to denote Binomial coefficients may be confusing to readers from different backgrounds.
In contrast, there is only one representation of the \binom notation, which is friendlier.