As a ballroom dancer who has competed in American Rhythm (where both ECS and Mambo are done)and performed and social danced the street styles of the dances as well as competed in the Country Western form, I can tell you what I teach to be the difference. Swing, in it's basic form is two triples and a rock step, social leaders need to be aware of their followers and the music, followers need to be aware of their leaders and their balance. Once you get beyond that, East Coast Swing was created by the ballroom system, as was Jive. They are versions of swing that fit the techniques of Latin and Rhythm styles. Lindy Hop is the original swing and has no hip action, a prone posture, and a downward bounce. West Coast Swing is a derivative, like East Coast and Jive, of Lindy Hop. It comes from California, where movie directors wanted a version of swing that didn't rotate so much. That way you didn't get a camera full of someone's behind. The current style of West Coast swing has a lot to do with the music it's danced to.
Salsa/Mambo is pretty similar to the ballroom style. The Ballroom style tends to be more presentational and with straighter legs, but it should still have the same elements of hip action and connection.
Hope that was helpful
Solomon
www.solomonsinger.com