The words peace, piece sound the same but have different meanings and spellings. Why do peace, piece sound the same even though they are completely different words?
The answer is simple: peace, piece are homophones of the English language.
The absence of war or other hostilities.
An agreement or a treaty to end hostilities.
Freedom from quarrels and disagreement; harmonious relations: roommates living in peace with each other.
Public security and order: was arrested for disturbing the peace.
A coin: a ten-cent piece.
A thing considered as a unit or an element of a larger thing, quantity, or class; a portion: a piece of string.
A portion or part that has been separated from a whole: a piece of cake.
An object that is one member of a group or class: a piece of furniture.
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").