When they have an imperative meaning like "let's go" both verbs can be interchangeable and can have the same meaning (see Sergio Romero's answer to see the difference). The question you may ask is why there are 2 ways of saying that and it's because the verb "ir" is used many times in a pronominal way as "irse" with the same meaning. So we have:
- Vámonos = Vamos + nos ("vamos" loses the final "s" when "nos" is added)
Note that in the second case "nos" is placed after the verb "vamos" because it has an imperative meaning, so "vámonos" can't mean something like "we go"(if you want to say "we go" you should say "nos vamos").
If you go to the conjugation table of the verb "ir" you'll see that "vamos" is the present of the verb "ir" (so it can also mean "we go" instead of "let's go" depending on the context). So you may wonder why a present can have an imperative meaning. The imperative in Spanish doesn't have a proper form for "nosotros", so when that happens the "imperative" is formed with the present of the subjunctive mood, for example:
seamos buenos (verb SER)
cantemos una canción (verb CANTAR)
So following the same rule you may think it should be "vayamos/ vayámonos". You may say that, though currently that form is only used in literature. But verb IR is a bit tricky in this case. It's done in a different way because of historical reasons (as it's explained in the 2nd point of this link from RAE). In the past, the present of subjunctive for "nosotros" could be both "vamos" and "vayamos". Now "vayamos" is the only subjunctive form, but "vamos" is much more frequent when it has an imperative meaning (as inherited from the past). Here we have what it says:
La forma vamos es hoy la primera persona del plural del presente de indicativo pero en el español medieval y clásico era, alternando con vayamos, forma de primera persona del plural del presente de subjuntivo. Como resto de su antiguo valor de subjuntivo, la forma vamos se emplea, con más frecuencia que vayamos, con finalidad exhortativa: «Vamos, Johnny, vamos a casa que es tarde»
La forma de subjuntivo vayamos, con este sentido, ha quedado casi relegada a la lengua literaria: «Vayámonos de aquí». Lo que no debe hacerse en ningún caso es emplear hoy la forma vamos, en lugar de vayamos, en contextos que exigen subjuntivo y sin que exista, en el enunciado, intención exhortativa.