The Western versions of the game are actually heavily edited versions of what Japan would get. While the first two games were never edited that hard, maybe a name change or two, or a panty shot removed, they were still mostly identical no matter which version you played. Here, the two versions are so very different that they're both worth playing through just for comparion's sake.
Here, the game starts with the redundantly-named Wood Oak City being devastated by a nuclear weapon containing a radioactive element called Rakushin. The Syndicate ends up stealing a large quantity of Rakushin, and then goes on to kidnap the high-ranking General Petrov. While the original Petrov is against the use of Rakushin, Mr. X sends in Shiva as an impostor in an attempt to start a war so that the Syndicate can smuggle the Rakushin they stole. And you know that fancy looking government building you go to for the 'bad' Stage 7, the one that looks like the White House? Surprise, it turns out that it's actually the White House. No matter which of the endings you get, the heroes manage to expose the fake general and stop a full scale war from breaking out.
So, there's absolutely nothing in BK3 about robotic duplicates, and nothing in SOR3 about any nuclear weapons. It's hard to say for sure why the plot ended up being changed so heavily. Maybe Sega of America thought there were too many political overtones with the main cast trying to stop a war, or maybe they thought the old story was too complicated. The new story isn't really that bad on its own, but the original plot means that the stakes are a little higher than some guy trying to take over some city for the third time.
The most notable change is that the difficulty isn't nearly as brutal as it gets in the Western versions. Since enemies aren't quite so aggressive or take as much damage, the difficulty level is a little more in line with SOR2. The other biggest change to the gameplay is that there's a new mid-boss in the middle of Stage 1, who can also be unlocked as a secret character. After taking out a few of his mooks he sends in, you'll end up in a fight with Ash, who even comes with his own, very odd theme. Ash is...well, he's a very uncomfortable stereotype, given his outfit and mannerisms. Despite his less than threatening appearance, however, he's actually really strong when you use him as a playable character, with his basic combo being able to take out an entire life bar. Once you get past the initial awkwardness of playing as him, however, he's not all that fun, as he's got the smallest moveset in the entire game. If you really want to play as him, there's enough of his code in SOR3 that you can play as him just fine with a Game Genie.
There's also a lot of minor visual and audio changes to the game, such as all of the characters getting a different palette for no discernable reason. Some of the female enemies have also been changed so that they're wearing a little more, as well. Some of the voice clips in the game have also been re-recorded so that they're in English, which is a little odd, considering that they never bothered to in SOR2. It's not that the Western version is that bad, unless you're the sort of person who can't stand needless censorship in your games. The one biggest problem with SOR3 is that it's just far, far too difficult, which makes BK3 the definite version of the game for sure. Thankfully, there's a translation patch for Bare Knuckle 3, meaning that it's perfectly possible to play both versions for yourself.
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