Plane surveying assumes the Earth's surface is flat and is used for small areas less than 260 square km, while geodetic surveying accounts for the Earth's curvature and is used for larger areas. Plane surveying forms plane triangles and uses simpler instruments, whereas geodetic surveying forms spherical triangles with more precise instruments like GPS. The main difference is that plane surveying ignores Earth's curvature while geodetic surveying considers it.