In French, "ventre" and "estomac" can both translate as "stomach", albeit with certain nuances:
"ventre": a more general, all-encompassing term referring to your (exterior) abdomen which contains your stomach, bowels etc, and a rather casual word.
"estomac": a part-specific term referring to your stomach as an organ, and a bit on the formal side, often heard on a medical context.
I assume that in this respect, "живот" corresponds to "ventre" and "желудок" corresponds to "estomac".
But here's the contradiction I've realised. When I say the following in conversation in a casual, non-medical context, I use "желудок":
1) На полный желудок и думается яснее.
... whereas in French only "ventre" fits this context:
1) Il n’y a rien de tel que d’avoir le ventre plein pour se tranquilliser l’esprit.
"Avoir le ventre plein" is something you'd say after having a hearty meal, while "avoir l'estomac plein" is more like what you'll see in a health-related or medical article. So, at least theoretically, I should be using "живот" here instead.
On a similar note, another contradictory case I notice:
2) И у меня живот вот-вот заурчит.
I use "живот" here, while in French "estomac" more fits this context:
2) Mon estomac ne va pas tarder à crier famine non plus.
... although at least if its French equivalent is anything to go by, I should be using "желудок" in this case.
So on the surface, "живот" corresponds to "ventre" and "желудок" corresponds to "estomac". But in practice, when it comes to using them in a sentence, as part of an expression, I notice it can work the other way around. So the distinction between "живот" and "желудок" in Russian doesn't seem as clear-cut as I'd expected it to be.