my question particularly applies to network reconstruction
Correlation measures the linear relationship (Pearson's correlation) or monotonic relationship (Spearman's correlation) between two variables, X and Y.
Mutual information is more general and measures the reduction of uncertainty in Y after observing X. It is the KL distance between the joint density and the product of the individual densities. So MI can measure non-monotonic relationships and other more complicated relationships.
To add to Rob's answer ... with respect to reverse engineering a network, MI may be preferred over correlation when you want to extract causal rather than associative links in your network. Correlation networks are purely associative. But for MI, you need more data and computing power.