According to my study, a plank is a long narrow piece of wood and a board is a thin flat piece of wood.
Now if a piece of wood that is long and narrow and thin, is it a board?
and if a piece of wood that is thin and flat and narrow, is it a plank?
are a plank and a board types of lumber?
A plank is a piece of lumber, it is a long, narrow, flat, piece. It has no particular purpose, and so could be used for anything where a piece of wood of that shape is needed. You could use several to make a floor, or to stop a door from being opened. You could stick one out the side of your pirate boat and force people to walk off it. A plank should be thick enough to support at least its own weight. Very thin strips of wood that have no structural use aren't planks.
A board is a flat and thin piece of wood (or other material) it might not be long and narrow, it could be squarish or even round. It often has a particular purpose, and so combines with other words. A plank used to make a floor is a now a floorboard. A board with words written on it is a signboard. A board with a checker pattern for playing games is a chessboard. Paint it black and its a blackboard (or chalkboard).
I believe the two words are close synonyms, and the context probably has most to do with which one is used.
Planks are used among other things to provide temporary bridging, loose walking surfaces - eg with scaffolding, and pirates "walk the plank". Planks are always long and narrow.
Boards come in different shapes and sizes, and can be used for many things including writing surfaces - blackboards, flooring - floorboards (actors "tread the boards"), shuttering - the shop is "boarded-up", notices - notice boards, etc.
