Asi que/por lo tanto

Asi que/por lo tanto
 

As a non-native Spanish speaker, I am a little uncomfortable about there terms, and often use pués instead. Dictionaries state they both mean so or therefore. Are there fine differences in the two terms?

I don't understand what pues (other thank as a expletive -a filler word-) has to do with this, but por lo tanto implies a logical consecuence of the previous phrase, while así que points to a sequence of facts.

En la caja fuerte había 10.000$ y ahora han desaparecido. Nadie la forzó, por lo tanto alguien que sabe la combinación es el responsable.

El abono mensual se fue al doble, así que me desuscribí.

You can swap them, but it will be overkilling in the second case.

Tengo hambre, así que voy a comer algo.

This one with por lo tanto would sound pretty ridiculous, because eating something is a justified decision, but not an intellectual conclusión.

I will explain to you my use of pués as I sometimes hear it.
Tengo hambre.
Pués, vamos a comer.

I now assume that asi que can best be used here. Does the use of pués in the above example sound wrong to you?

It sounds regional to me. An I continue to perceive it as an expletive (in grammar: a filler word), because "Tengo hambre. Vamos a comer" sounds just fine.

Other thing is in a dialogue

-Tengo hambre
-Pues, ¡vayamos a comer!

here pues means entonces, though you may hear

-Pues entonces, ¡ vayamos a comer!

with pues becoming again an expletive.

The use of pues and vamos sounds regional to me, and carrying a slightly different meaning

Vayamos a comer = Let's go (to the diner)
Vamos a comer = (To the diner) we go [Like in "there we go"]