The words rout, route sound the same but have different meanings and spellings. Why do rout, route sound the same even though they are completely different words?
The answer is simple: rout, route are homophones of the English language.
An overwhelming defeat.
A disorderly retreat or flight following defeat.
A disorderly crowd of people; a mob.
People of the lowest class; rabble.
A road, course, or way for travel from one place to another.
A highway.
A customary line of travel. See Synonyms at way.
A fixed course or territory assigned to a salesperson or delivery person.
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").