Think of the difference like of any other statistic that you are collecting. These differences are just some values that you have recorded. You calculate their mean and standard deviation to understand how they are spread (for example, in relation to 0) in a unit-independent fashion. 

The usefulness of the SD is in its popularity -- if you tell me your mean and SD, I have a better understanding of the data then if you tell me the results of a TOST that I would have to look up first.

Also, I'm not sure how the difference and its SD relate to a correlation coefficient (I assume that you refer to the correlation between two variables for which you also calculate the pairwise differences). These are two very different things. You can have no correlation but a significant MD, or vice versa, or both, or none.

By the way, do you mean the standard deviation of the mean difference or standard deviation of the difference?