The inverse of covariance matrix can be used to find the conditional independencies among variables. This inverse is more sensitive to changes then the correlation matrix. As a result two samples with very similar correlation matrices and standard deviations, may still point to different conditional independencies.
Intuitively it is easy to see why this must be the case. The covariance matrix is the Hadamard product (multiplication by entry) of the correlation matrix and the outer product of the standard deviation. An outer product by definition has rank 1, all rows are linearly dependent. A very small change in one entry in each row would lead to an inversible but very ill-behaved matrix, its inverse would be very dependent on small changes. That multiplying a matrix with a very ill-behaved matrix would make it more sensitive to perturbations seems reasonable.
Can it be expressed formally, for example using the condition number, that the product of a matrix with a rank deficient matrix has a larger condition number than the original matrix?
Or more generally, can it be shown that if |A||A^-1| < |B||B^-1| then |A||A^-1| < |(AB)| |(AB)^1| (Where |x| is a norm.)