The words coarser, courser sound the same but have different meanings and spellings. Why do coarser, courser sound the same even though they are completely different words?
The answer is simple: coarser, courser are homophones of the English language.
comparative form of coarse: more coarse
A dog trained for coursing.
A swift horse; a charger.
Definitions from Wiktionary, Creative Commons Attribution/Share-Alike License, 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").