Created in DuPont's lab back in 1958, it was named after its physical ability to "expand." Spandex is a synthetic fiber stronger and more durable than natural rubber.
It can be stretched up to 6x its original length and return back to its original shape. If the average newborn (20") were stretched 600%, it would stand as tall as a basketball rim.
Lycra has been DuPont's branded name since 1962. Think of it as "Kleenex" for tissues or "Dri-FIT" for microfiber polyester fabric.
Though you may find latex in elastic waist bands, it is generally not used in clothing because of the high potential for allergic reactions. Another difference is that Latex is made using natural rubber.
Elastane is a more commonly used phrase in Europe and other parts of the world. However, they are referring to the same polyurethane material.