A skim of the documentation reveals much similarity. They both create or enable an alternate representation of data that sits somewhere else without having to actually create a new image. Examples include creating a single seamless image from many tiles without actually creating that single image, or mix-and-matching channels or selectively removing value ranges from the result, and more.
How far does that similarity extend? Does an RMD enable things that VRT can't? What are they? What can one do with VRTs that you can't with RMDs?
In short, assuming one has equal access to and comfort using both tool chains, when should a person use Raster Mosaic Datasets and when are Virtual Raster Tables the order of the day?