"Vamos" is a verbal form of the verb "ir" that means "to go" and "vámonos" is a verbal form of the verb "irse" that means "to leave". So when you use "vamos" it's like saying "let's go" and the person you're talking to knows where you two go to. I mean that for example if you two are in a bar and you're going to go to the cinema and it's time to go, you can say only "vamos" because it's understood that you're going to the cinema.
Instead, when you say "vámonos", you're saying to your partner to go out from the place where you two are. For example, if you two are in a bar but you don't like the place too much and want to leave, you would say "vámonos". Maybe you don't have a plan to go to other place, you only want to leave the bar. Or maybe you have the plan to go to other place and you say "vámonos" and mean both things, leave the place and go to another.