In the US stomach generally is used for both the interior organ and the exterior abdomen.
- The child has a stomach ache
-All girls want to have a flat stomach
-The man was punched in the stomach.
-Don't drink on an empty stomach.
-I think this dress makes my stomach look fat.
-The stomach is part of the digestive system.
-Some people like to sleep on their stomachs.
Belly is also used but often in a negative way.
-"Look at the beer belly on that man!" (A fat protruding abdomen often blamed on excessive beer consumption)
-Stop belly-aching! (Para de reclamar!)
-Eating wheat can cause belly fat.
Abdomen is used in the context of fitness. (often abbreviated as abs)
-The physical therapist gave me some abdominal exercises to do at home
-Athlete: The coach had us working our abs at practice today.
There is one use of belly that is universal and common: belly button (umbilico? Umbigo? Which is it? ) It would be very funny to hear "stomach button." ha ha ha.
In some places belly ache is used instead of stomach ache, especially with children. Although where I live we always say stomach ache, not belly ache.
There also is the phrase "belly flop" which is when you dive into the water flat on your stomach.
The boy did a belly flop into the pool and splashed everyone.
Also "belly up" which means to go out of business.
The company went belly up when the founder was arrested on tax fraud.
"Belly up" also means to approach a bar to get a drink. (Not common or polite so don't say it.)
In conclusion, belly can be used interchangeably with stomach. I always say stomach except when referring to the belly button or to belly fat.