The words rhyme, rime sound the same but have different meanings and spellings. Why do rhyme, rime sound the same even though they are completely different words?
The answer is simple: rhyme, rime are homophones of the English language.
To make use of rhymes in composing verse.
Correspondence of terminal sounds of words or of lines of verse.
A poem or verse having a regular correspondence of sounds, especially at the ends of lines.
Poetry or verse of this kind.
A coating of ice, as on grass and trees, formed when extremely cold water droplets freeze almost instantly on a cold surface.
A coating, as of mud or slime, likened to a frosty film: "A meal couldn't leave us feeling really full unless it laid down a rime of fat globules in our mouths and stomachs” ( James Fallows).
To cover with or as if with frost or ice: "heavy [shoes] rimed with mud and cement ... from the building site” ( Seamus Deane).
Variant of rhyme.
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").