English isn't my first language, so I'm not 100% sure, but I found this on the net;
"Ghost is the general word for the appearance of a dead person in the semblance of his living form. Phantom can apply to any manifestation, pleasant or not, any illusion or idea, such as "the phantom of a future utopia"."
(I slightly disagree with this one, since ghost/phantom/spirit are synonyms, including illusion)
I would say, they are more or less alike, though there are some differences in where they get used. Also,"ghost" is most common, it applies more to daily life, while "phantom" sounds more poetic and therefore used less.