The words purest, purist sound the same but have different meanings and spellings. Why do purest, purist sound the same even though they are completely different words?
The answer is simple: purest, purist are homophones of the English language.
superlative form of pure: most pure.
having the greatest purity
One who practices or urges strict correctness, especially in the use of words.
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").