The words hoarse, horse sound the same but have different meanings and spellings. Why do hoarse, horse sound the same even though they are completely different words?
The answer is simple: hoarse, horse are homophones of the English language.
Rough or grating in sound: a hoarse cry.
Having or characterized by a husky, grating voice: yelled ourselves hoarse.
Drawn or operated by a horse.
A large hoofed mammal (Equus caballus) having a short-haired coat, a long mane, and a long tail, domesticated since ancient times and used for riding and for drawing or carrying loads.
An adult male horse; a stallion.
Any of various equine mammals, such as the wild Asian species E. przewalskii or certain extinct forms related ancestrally to the modern horse.
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").