The Mambo dance that was invented by Perez Prado and was popular in the 1940s and 50s in Cuba, Mexico City, and New York is completely different from the modern dance that New Yorkers now call 'Mambo', which is also known as Salsa "on 2".
This new rhythm was harder to teach (2,3,4 hold 1) so many studios are opting for the 1,2,3 hold 4 as Americans are used to stepping and counting the "One" first.
Basically Mambo is a faster single rhythm and Cha Cha is the slower triple rhythm dance.
The mambo replaced the Rumba in popularity and gave birth to the Cha-Cha (or triple mambo).
The word Salsa today is used to describe the music and the dance, a modern term for Latin if you will. However the dance known as Salsa is really just a slowed down MUTATED Mambo and usually breaking on count One, rather than the Mambo's "break on Two" plus the emphasis is on dance patterns.