The earth's shape is approximately a sphere, but as you gentlemen point out, not exactly so. For navigating ships, this is not an issue, but for mapping activities and surveying it does make a difference.

Navigators assume the earth to be perfect sphere.

The first approximation correction is a slightly flattened-at-the-poles sphere, or an ellipsoid of revolution. Several variations of this exist, using slightly different ratios of polar to equatorial diameter.

But even this is wrong for close work, so slightly different spheroids are chosen for different parts of the world, where they are selected for better fits in those particular areas. These are usually continental-sized areas, like the Everest (he was a surveyor) in India, or the Clarke Spheroid in N. America. If we mapped between these areas the maps would not agree along the boundaries, but we just dump the error into the oceans where no one will notice it.

Today, we not only try to take into account the true geometrical shape of the earth, but its mass distribution as well. A level or plumb bob on a surveyor's instrument does not point to the geometrical center of the planet, but to it's center of mass, modified by rotation, Coriolis, etc.. Not the same thing in our case.

Satellite observations and GPS have allowed us to come up with even more refined observations, so today Clarke 1867 has been largely replaced by the North American Datum of 1983, and its many variations. None of these are the true shape of the earth, because the earth's shape deviates randomly from an ellipsoid of revolution in different places, and the ellipsoid is always chosen to be a true geometrical shape which can be modeled by an equation, rather than a grid of measured elevations. BTW, by "shape" I mean "average shape", I'm not talking about displacements due to terrain relief. So different places still use different spheroids which are convenient for their geographical corrections. Lat/Lon locations referenced to different spheroids usually differ from each other by several hundred feet, in x, y and not quite that much in z, which is no big deal for a ship's captain, but a very big deal if you're trying to land a plane in a fog.

That's why your GPS has all those annoying options which allow you to program differerent spheroids, and why it is important you pick the correct spheroid to match the map you are using to fully exploit the accuracy of your receiver. Finding Hawaii is one thing, if you want your GPS to warn you when you're dragging anchor, you're going to need more accuracy.