In my experience, a more complete answer to this question lies in defining the difference between a DEM, DTM and a DSM. A DTM is NOT a generic name covering both DEMs and DSMs. So...
A DEM is a 'bare earth' elevation model, unmodified from its original data source (such as lidar, ifsar, or an autocorrelated photogrammetric surface) which is supposedly free of vegetation, buildings, and other 'non ground' objects.
A DSM is an elevation model that includes the tops of buildings, trees, powerlines, and any other objects. Commonly this is seen as a canopy model and only 'sees' ground where there is nothing else overtop of it.
A DTM is effectively a DEM that has been augmented by elements such as breaklines and observations other than the original data to correct for artifacts produced by using only the original data. This is often done by using photogrammetrically derived linework introduced into a DEM surface. An example is hydro-flattening commonly seen in elevation models done to FEMA specifications
Incidentally, a DEM is far cheaper to produce an a DTM.