Measuring distance between two points using digital elevation model (DEM) and ArcMap

Any GIS can give you the distance between two points as long as you have a map projection defined. ArcGIS allows some GIS function in a geographic coordinate system which doesn't have a map projection defined, so it won't give you an answer using the measure tool between two points (it also doesn't tell you why there's no answer). You should know which "map projection" you are using because your points and your 'data frame' might be in different coordinate systems (that's OK if you're good with the measurement error to convert between the two systems). Points and the lines between them (even if it's temporary like the line the measurement tool makes) are examples of vector data. DEMs are raster data. If you are writing something to calculate the distance between two cells in the DEM, you use raster tools; there are various vector tools to get the distance between points.