"Pass gas" sounds more polite than "fart". "Fart" isn't a rude word, but it sounds more direct. It's similar to "die" and "pass away" in that we prefer to refer to unpleasant concepts with less direct words. "Arse" is British English and "ass" is American English. Keep in mind that in British English, the "r" is silent and "ar" is used to indicate a long "ah" sound, thus they have the same pronunciation (except in slightly different accents obviously). We normally use British English spelling in Australia, but I've only ever seen the American spelling of the word here for some reason. "Ass" is also an old-fashioned word for "donkey" in all countries.

There's no difference, but use "pass gas." It's safer. "Fart" was considered unacceptably vulgar in the 1950s. In the United States nowadays it is ALMOST standard English--but not quite. It is informal. When I was recovering from surgery, the nurses asked me "Have you passed gas yet;" I can't imagine a nurse asking a patient "have you farted yet?" I see that the American Heritage dictionary still designates the word as "vulgar slang." "To break wind" is another synonym, but it's rather old-fashioned. The correct medical term for intestinal gas is "flatus," and farting a lot is "flatulence."