The words wade, weighed sound the same but have different meanings and spellings. Why do wade, weighed sound the same even though they are completely different words?
The answer is simple: wade, weighed are homophones of the English language.
To walk in or through water or something else that similarly impedes normal movement.
To make one's way arduously: waded through a boring report.
To cross or pass through (water, for example) with difficulty: wade a swift creek.
The act or an instance of wading.
Simple past tense and past participle of weigh.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition, from Wiktionary, Creative Commons Attribution/Share-Alike License 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").