The words whither, wither sound the same but have different meanings and spellings. Why do whither, wither sound the same even though they are completely different words?
The answer is simple: whither, wither are homophones of the English language.
To what place, result, or condition: Whither are we wandering?
To which specified place or position: landed on the shores whither the storm had tossed them.
To whatever place, result, or condition: "Whither thou goest, I will go” ( Ruth 1:16).
To dry up or shrivel from or as if from loss of moisture.
To lose freshness; droop.
To cause to shrivel or fade.
To render speechless or incapable of action; stun: The teacher withered the noisy student with a glance.
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").