The words puer, pure sound the same but have different meanings and spellings. Why do puer, pure sound the same even though they are completely different words?
The answer is simple: puer, pure are homophones of the English language.
Canine faeces used as a lye in tanning.
Theoretical: pure science.
Having a homogeneous or uniform composition; not mixed: pure oxygen.
Free from adulterants or impurities: pure chocolate.
Free of dirt, defilement, or pollution: "A memory without blot or contamination must be . . . an inexhaustible source of pure refreshment” ( Charlotte Brontë).
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").