To clarify for anyone reading, the difference is not constant vs variable magnitude of g, it's constant vs variable direction of g. Gravity is a vector pointing at the (weighted by inverse of distance squared) centre of mass, so once you are operating on a global scale you have to account for the direction of g changing as you move around.

The force always pointing towards a single point is what causes an ellipse to form. If the force always pointed down it would be a parabola, and over short distances on the surface this is a really good approximation.