The words groan, grown sound the same but have different meanings and spellings. Why do groan, grown sound the same even though they are completely different words?
The answer is simple: groan, grown are homophones of the English language.
To voice a deep, inarticulate sound, as of pain, grief, or displeasure.
To make a sound expressive of stress or strain: floorboards groaning.
To utter or express with groans or a groan.
The sound made in groaning.
Past participle of grow.
Having full growth; mature: a grown woman.
Produced or cultivated in a certain way or place: locally grown produce.
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").