I would think the simplist explanation is that the Mushroom World is either a large country or literally the entire world that the Mario characters live in and that the Mushroom Kingdom is one of many kingdoms within it. Like New York City is part of New York State.
SMB 3 says Mushroom World, not Mushroom Kingdom, so I would just assume there would be some differentiation. Surely if it took place in the Mushroom Kingdom they would have just said "the lands of the Mushroom Kingdom". Assuming that each royal family has their own kingdom is a much more logical assumption than to assume some big empire where each royal family answers to some greater kingdom that rules over all of them.
This whole discussion demonstrates how annoying it is that Nintendo is so disinterested in having some sort of consistent canon. What other company just throws this stuff out without any real explanation and leaves their fans to guess what's what? If I was a game designer the world building would be part of the appeal and I sure as hell would try to make sure it all made sense.
It's only frustrating when you take it seriously like that. It wasn't until I was between 18 or 21 when I finally started to realize that continuity in games is pointless and any story or setting that goes along with it should be taken at face value. You Ian specifically made me realize game writers are "failed movie and book writers," there is never anything to count on. At least Nintendo isn't trying to put all the Mario titles together, explaining everything like there is true meaning and value behind it. It's the developers that try to do it with their epic story lines and explanations that look even more foolish.
I love the people who follow Kingdom Hearts, Zelda, Sonic(used to be me), Metal Gear religiously. So much convoluted bull crap yet the true rubes go "that's deep" and just buy into the whole thing. Don't get me wrong I love a decent story in games, and I really love when everything is clear and explained, but what developer out there cares about that stuff as much as the fans and is even slightly aware of every detail like the fans to keep it consistent? None of em.
So to answer the question, especially with Mario, who cares? It's probably best not to think about it when you don't have Miyamoto himself there with you answering questions. Still I'm certainly guilty of this, I have an emotional investment in the Shell VS Saddle debate.
Now then, the true moral to this story is you all need to read One Piece. It's extremely consistent the writer is amazing and knows more about his story then the fans. It's the only thing any one can believe in now a'days.