When and where should we use Geodetic Densify over Densify and vice versa?
Densify - 2D Cartesian based on the data's coordinate system. It had better be projected, unless the data has a small extent (and possibly not at mid- to high-latitudes). It's fine for densifying lines or polygons for a city, county, or small state when the data is in a reasonable projected coordinate reference system. Never use it if the data is in a geographic coordinate reference system.
Use Densify when you want to maintain the "straightness" of the lines--like streets or parcel boundaries, things like that. It could include cases where you have lat/lon, but you want to densify along the latitude line--think the US-Canadian border in the west
Geodetic Densify - Densifies on the ellipsoid surface using any of four options, so more accurate for larger extents or at mid- to high-latitudes. The calculations are performed on the ellipsoid associated with the input data's coordinate system.
Before and after
Densify are the same rendered in pseudo-Plate Carree projection. The picture looks the same--straight line segments because the world is treated as if it's flat:
Geodetic Densify the lines are curved when displayed in pseudo-Plate Carree:
(pictures cropped from Esri help topic)