The sphere is actually very good approximation, but as a tribute to our precision we must consider departures from a sphere for any map of an area smaller than a continent, and anytime we need precise coordinates such as rocket launches (where we also need to know the gravity field), property surveys, and locating things like oil wells.
Spheroid: adequate for small scale maps of the entire world or a hemisphere.
Ellipsoid: more complex mathematics, defined as part of the horizontal datum
The Geoid (gravity equipotential surface, mean sea level) has many oceanographic applications, and is how we currently have complete bathymetric coverage in the oceans.
Last revision 11/13/2017