All four words describe land with plant growth, but they differ in emphasis. Grassy focuses specifically on grass covering the ground — it tells you what the surface is made of, not just its colour. Green is the broadest term: it can describe grass, trees, leaves, or even a general look (e.g. 'green hills'), so it is less precise. Verdant is a more formal and poetic word that emphasises rich, healthy vegetation — you would use it in nature writing rather than everyday speech. Lush stresses thickness and abundance: a 'lush lawn' has very dense, healthy grass, whereas a 'grassy field' could be thin or patchy. For everyday descriptions of a grass-covered surface, grassy is the most direct and practical choice.
Grassy comes from the Old English word græs, meaning 'grass', with the suffix -y added to mean 'covered with' or 'full of'. English has used this combination since the Middle Ages to describe land that is naturally covered in grass rather than bare or cultivated.