RMS, every organ does have a distinct sound to it. I believe it's a combination of factors including type of leslie, age of leslie, age of tubes (tubes have to be broken in), age of organ, whether or not the organ's tone generator has old or new capacitors, etc....
The New B-3 actually has an option where you set the sound to emulate a clean, "perfect" B-3 or other levels of "vintage-ness." It's not only on the New B-3, either. Many of the other synths have similar options.