The words coffer, cougher sound the same but have different meanings and spellings. Why do coffer, cougher sound the same even though they are completely different words?
The answer is simple: coffer, cougher are homophones of the English language.
Financial resources; funds.
A treasury: stole money from the union coffers.
Architecture A decorative sunken panel in a ceiling, dome, soffit, or vault.
A person who coughs.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition, from Wiktionary, Creative Commons Attribution/Share-Alike License 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").