Both of these words are used to mean let's go and they are both nosotros command forms of the verb ir. The difference is that vámonos is the command form with a pronoun attached.

What Vamos mean?

One Spanish term that caught on with English speakers was vamos, which means "let's go." Cowpokes and dudes alike adopted the word, at first using a range of spellings and pronunciations that varied considerably in their proximity to the original Spanish form. ... Get Word of the Day delivered to your inbox!

Which is correct Vamos or vamanos?

Usually "vámonos" would be used in a context in which you are leaving FROM a place, something like "Vámonos de este lugar" - "Let's go from this place". Regarding "vamos" would be used in a context in which you are going TO a place, something like "Vamos al cine" - "Let's go to the movies".