The words auger, augur sound the same but have different meanings and spellings. Why do auger, augur sound the same even though they are completely different words?
The answer is simple: auger, augur are homophones of the English language.
Any of various hand tools, typically having a threaded shank and cross handle, used for boring holes in wood or ice.
A drill bit.
A machine having a rotating helical shaft for boring into the earth.
A rotating helical shaft used to convey material, as in a snow blower.
One of a group of ancient Roman religious officials who foretold events by observing and interpreting signs and omens.
A seer or prophet; a soothsayer.
To predict, especially from signs or omens; foretell. See Synonyms at foretell.
To serve as an omen of; betoken: trends that augur change in society.
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").