Both have several meanings but they round up to mean flatness. But their usage is different and one has to be careful when to use what. The difference in their usage is explained below.
Plain as an adjective means "easily understood," "undistinguished," or "unadorned".
His meaning was plain to all.
The plain dress suited the gravity of the occasion.
As an adverb, ,plain means "clearly and simply".
She's just plain foolish.
As a noun, plain is a flat area of land.
The vast plain seemed to go on forever.
As a noun, plane has a number of different meanings. It most commonly refers to an airplane, but is also used in mathematics and fine arts and is a tool used to shave wood. Plane usually means a “flat and level", or also mean a “two dimensional surface".
When does your plane take off?
They have raised high above the plane.
Examples of plain:
He said that he don't agree with this quite plainly.
She said it seemed plain to her.
He cannot run away from plain truth.
Examples of plane:
He ordered the worker to plane a few inches high.
She is nervous as it is her first trip on a plane.
Their car skidded and planed straight into the garden