The words planar, planer sound the same but have different meanings and spellings. Why do planar, planer sound the same even though they are completely different words?
The answer is simple: planar, planer are homophones of the English language.
Of, relating to, or situated in a plane.
Flat: a planar surface.
Having a two-dimensional characteristic.
One that planes, especially a machine tool that is used to smooth or finish the surfaces of wood or metal.
Printing A smooth block of wood used to level a form of type.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.
If they are spelled the same then they are also homographs (and homonyms); if they are spelled differently then they are also heterographs (literally "different writing").