Frank,
The difference is subtle, but 'vamos' means 'let's go (somewhere)', and 'vámonos' means 'let's go' (as in, 'let's leave').
So consider these two situations:
¿Vamos al cine?
¡Sí! ¡Vamos!
Versus:
¿Ya nos vamos? (Let's get out of here?)
¡Sí! ¡Vámonos!