In what situations is covariance preferred to correlation? If the magnitude of covariance doesn't really convey any reliable information, and correlation provides us the same information but also gives the degree of dependence, then why use covariance at all ? 
 A: I would say that covariance gives you additional information, which is the scale of the input and output standard deviation since $\sigma_{xy}=\sigma_{x}\sigma_{y}\rho_{xy}$ where the terms denote the covariance, the standard deviations, and the correlation, respectively.
On example, where the covariance is needed is modeling data with a multivariate Gaussian distribution $$p(\mathbf x) = \frac{1}{\sqrt{|2\pi\Sigma|}}\exp\left(-\frac{1}{2}(\mathbf x - \mu)^\top \Sigma^{-1}(\mathbf x - \mu)\right).$$
Here, the data covariance is the maximum likelihood estimator of the covariance of the Gaussian. A direct consequence of that is the formula for the conditional mean $$\mathbf x_2 = \Sigma_{21}\Sigma^{-1}_{11}(\mathbf x_1 - \mu_1) + \mu_2$$ where $\mathbf x_1$ and $\mathbf x_2$ denote a partition of the vector $\mathbf x$. The formula basically means that you predict parts of $\mathbf x$ from another part. In case of Gaussian data, this is the optimal linear predictor for least squares error.
