This work considers the asymptotic behavior of the distance between two sample covariance matrices (SCM). A general result is provided for a class of functionals that can be expressed as sums of traces of functions that are separately applied to each covariance matrix. In particular, this class includes very conventional metrics, such as the Euclidean distance or Jeffrery's divergence, as well as a number of other more sophisticated distances recently derived from Riemannian geometry considerations, such as the log-Euclidean metric. In particular, we analyze the asymptotic behavior of this class of functionals by establishing a central limit theorem that allows us to describe their asymptotic statistical law. In order to account for the fact that the sample sizes of two SCMs are of the same order of magnitude as their observation dimension, results are provided by assuming that these parameters grow to infinity while their quotients converge to fixed quantities. Numerical results illustrate how this type of result can be used in order to predict the performance of these metrics in practical machine learning algorithms, such as clustering of SCMs.