He's asking about 9x18. That's 9mm Makarov. Not 9x19 - 9mm Luger. I'm not sure, from his answer, if graehaven was aware of that, but RAJBCPA's answer tells me that he isn't.
9x18 shoots a slightly heavier bullet, with slightly more power. SLIGHTLY. Ammo used to be cheaper, but the Mak surplus and Russian stuff seems to have dried up, so the price difference might not be there anymore.
380 has more available "self defense"-type bullets and more self-defense loads than the Mak does.
Gun size - the smallest I'm aware of, in Mak, is a Polish pistol that is about the size of a Walther PP. In 380 they make quite little guns, these days.
I believe its popularity came more from the cheap guns and cheap surplus ammo, than any advantage it might have over the 380. Nowadays I think it is more personal preference than anything else.