According to most of the answers on What is the difference between DEM, DSM and DTM?, there is a difference between a DEM and a DTM.
In particular this answer seems to offer a clear explanation of the difference between a DEM and a DTM:
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.