The words climb, clime sound the same but have different meanings and spellings. Why do climb, clime sound the same even though they are completely different words?
The answer is simple: climb, clime are homophones of the English language.
To move upward on or mount, especially by using the hands and feet or the feet alone; ascend: climb a mountain; climbed the stairs.
To grow in an upward direction on or over: ivy climbing the walls.
To move oneself upward, especially by using the hands and feet.
To rise slowly, steadily, or effortfully; ascend. See Synonyms at rise.
Climate: in search of warmer climes.
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.