The words weather, wether, whether sound the same but have different meanings and spellings. Why do weather, wether, whether sound the same even though they are completely different words?
The answer is simple: weather, wether, whether are homophones of the English language.
The state of the atmosphere at a given time and place, with respect to variables such as temperature, moisture, wind velocity, and barometric pressure.
Adverse or destructive atmospheric conditions, such as high winds or heavy rain: encountered weather five miles out to sea.
The unpleasant or destructive effects of such atmospheric conditions: protected the house from the weather.
Changes of fortune: had known him in many weathers.
A castrated ram.
Used in indirect questions to introduce one alternative: We should find out whether the museum is open. See Usage Notes at doubt, if.
Used to introduce alternative possibilities: Whether she wins or whether she loses, this is her last tournament.
Either: He passed the test, whether by skill or luck.
Archaic Which: "We came in full View of a great Island or Continent, (for we knew not whether)” ( Jonathan Swift).
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").