I have read three different words with seemingly no difference in meaning: panza, barriga, and vientre. Are there regional differences in usage? Or do they differ in degrees of formality? I am not referring to estómago because it’s the external body part that I am talking about here.
"Vientre" is the most formal or technical. It's also quite ample: it can be used for the external as for the internal ("dolor de vientre" = "dolor de estómago") and also for the viscera. "Ir de vientre" is an euphemism (usual in medical conversation) for "defecate". It can even mean "womb" (but this is an archaism, from the latin: the christian prayer "Ave Maria" reads "Bendito el fruto de tu vientre").
"Barriga" is most informal-coloquial. It refers mainly to the visible exterior, specially when it's prominent (belly).
"Panza" is similar to "barriga", it's more used here (Argentina; I rarely say or hear "barriga"). In case of a stomach-ache, a kid would say "Me duele la panza".
For example: one might say "Juan tiene panza" as an equivalent "Juan tiene mucha panza" (John is fat - abdominally obese). One would never use "vientre" here.
In some places (Spain, I believe) the word "tripa" is used instead of "panza". That sounds strange in Argentina, where "tripa" means basically "viscera" or "intestine".
I would use «vientre» as the anatomical description of the area (both interior and exterior and without consideration of size), similar to «abdomen», while «barriga» and «panza» would be more used for a slightly rounded or greatly over-sized one from an external point of view.
For the official definitions:
So «vientre» does refer to the anatomical cavity (1.), the external anatomical region (3.), and the bowels in the abdomen (2.), although I have not seen much or would use this meaning.
So meanings 2. and 3. of «barriga» refer to a bulky part or a warp in some objects., while meanings 1. and 4. refer to anatomical «vientre» (meanings 1. and 2. respectively). Personally I would seldom use them in those anatomical contexts but mainly as in meaning 5.: “external region of the human body equivalent to abdomen, especially if bulky.” - the fact that is the fifth meaning might suggest a regional variation but I have probably heard it in rather this context than referring to the bowels or the anatomical cavity.
So «panza» does mean the bulky «barriga» or «vientre», as well as the bulky part of other objects. (and the cow's stomachs.)
Here in Chile we often refers as "Guata"; very informal and very common.
"Me duele la guata" (referring to stomach ache).
"Le pegó un combo en la guata" (referring to a punch in the stomach).