‘Plain’ is an adjective with many uses and shades of meaning around the ideas of ‘simple’, ‘direct’, and not fancy. For example, ‘plain yogurt’ has no flavoring. ‘Plain language’ means saying what you mean. It is also a noun used to describe a kind of topogrophy: a large, mostly flat region.
“The Great Plains of North America cover parts of 10 U.S. states and three Canadian Provinces.”
‘Plane’ means a flat, level surface (in mathematics or physics) or an airplane.
“We need to board the plane now.”
Plain refers to something in its natural form, unaffected by artificial modifications or editing. It advocates for simplicity.
"Unbelievable! The boss came to work with a plain T-shirt and shorts."
Plane could mean an airplane or an aeroplane which is obviously capable of flight. It also refers to an equipment used for smoothing pieces of wood by removing thin layers from the surface.
"I need to get on the plane headed for Canada in an hour!"