The words dour, dower sound the same but have different meanings and spellings. Why do dour, dower sound the same even though they are completely different words?
The answer is simple: dour, dower are homophones of the English language.
Marked by sternness or harshness; forbidding: a dour, self-sacrificing life.
Silently ill-humored; gloomy: the proverbially dour New England Puritan.
Sternly obstinate; unyielding: a dour determination.
The part or interest of a deceased man's real estate allotted by law to his widow for her lifetime. Also called dowry.
A natural endowment or gift; a dowry.
To give a dower to; endow.
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.