The words broom, brume sound the same but have different meanings and spellings. Why do broom, brume sound the same even though they are completely different words?
The answer is simple: broom, brume are homophones of the English language.
An implement used for sweeping, usually consisting of a bunch of twigs, straw, or bristles bound together and attached to a stick or handle.
Any of various Mediterranean shrubs of the genus Cytisus in the pea family, especially C. scoparius, having mostly compound leaves with three leaflets and showy, usually bright yellow flowers.
Any of several similar or related shrubs, especially in the genera Genista and Spartium.
To sweep with or as if with a broom.
Fog or mist.
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").