The words bow, bough sound the same but have different meanings and spellings. Why do bow, bough sound the same even though they are completely different words?
The answer is simple: bow, bough are homophones of the English language.
A tree branch, especially a large or main branch.
To cause to acquiesce; submit.
Nautical The front section of a ship or boat.
Nautical The oar or the person wielding the oar closest to the bow.
To bend or curve downward; stoop.
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.