They are synonyms. "Ghost" is the common, usual word. "Phantom" is rare and literary, so despite the cognate "fantasma," it is better to use "ghost." "Ghost" specifically means the visible spirit of a dead person. "Phantom" also has come mean a living person who keeps himself concealed or does things secretly--either serious things or practical jokes.

(P.S. There is also an English word, "phantasm," but it means "illusion" or "hallucination.") Examples: On Halloween, we will give candy to children dressed as skeletons and ghosts. (NEVER "phantoms.") "Do you believe in ghosts?" (NOT "phantoms.") "Before the phantom of False morning died..."--line from a famous 1868 poem, Edward FitzGerald's translation of the Rubaiyat of Omar Khayyam. It uses literary English. "Phantom" does not mean spirit of the dead. It means the sky appearing to brighten before it is actually dawn. "She was a Phantom of delight/When first she gleamed upon my sight... I saw her upon nearer view, A Spirit, yet a Woman too!" More 1800s poetry, this time by William Wordsworth. So, you could compliment a woman--in old-fashioned, courtly language--by saying "You are a phantom of delight," but you could NEVER say "you are a ghost of delight!"

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