Cultists of Happy Happyism appear in Mother 2 with the letters HH on their forehead (for "Happy Happyism") and lack a pom-pom at the end of their hoods. In Earthbound a pom pom was added, presumably to try to make the cultists look less like members of the Ku Klux Klan, and the HH was removed, possibly because "HH" on an already Klan-like sprite could be construed to mean "Heil Hitler".
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Threek changed to Threed in Earthbound. Like the cultists, this may also be to avoid any link to racist groups as Threek could be seen as Three-K or KKK.
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Nintendo of America's strict censorship policy resulted in changing all bars in the game to cafes.
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The signs on the hospitals in Onett are different. The MOTHER 2 hospital has a red cross on it, while the EarthBound version does not. The reason for this change was to avoid any legal problems with the Red Cross organization, since the Red Cross is protected by federal law. There's also a bunch of text written on the signs in Earthbound.
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The intro wording changed from Mother 2 to Earthbound(that being "The war against Gigyas"). The wording was modified because Mother was never released in the west, and it would be rather odd for an enemy never seen before to "strike back".
Note: I'm putting the last part because most Fans are still stuck on the whole Gigyas being a Fetus and stating that Gigyas can't possibly be the one from Mother as that messes up the whole Thoery.