Preliminary remark: Some other verbs can follow this path, in some contexts. For example, let's say my family and I are sitting down to eat, but we get involved in a heated discussion and the food gets forgotten. I can exhort or invite everyone to start eating, and continue the discussion calmly. There are several ways of doing this.
Comamos [subjunctive... imperative]
Vamos a comer [sort of future... sort of another type of imperative]
Comemos [present indicative... a suggestion]
The third approach might be clearer if I write it like this:
Comemos, ¿no? ¿Y seguimos discutiendo, pero con calma?
Now, to answer your question, the main reason we use "vámonos" instead of "vayámonos" typically is that it flows better in the context -- which generally calls for something short and punchy.
However, there are situations in which "vayamos" can be comfortably used. Examples:
Vayamos al tema de la investigación con células madre. | Let me turn to the issue of stem cell research. (Example taken from linguee.com)
No hay que quedarse con el estatu quo. Vayamos avanzando con nuestras metas. | We shouldn't stay stuck with the status quo. Let us move forward with our goals.
No nos vayamos todavía. Quiero quedarme otro rato. | Don't let's go yet. I want to stay a little longer.