nouns - Difference between "pain" and "ache"

What's the difference between pain and ache?

I often see the two words used (almost) interchangeably. At the same time the phrase "aches and pains" is pretty common, and seems to suggest that the two words aren't exact synonyms.

The dictionary definitions I've seen ("ache is a continuous dull pain") don't really help in understanding many of the usages I've come across, such as this one from the NHS:

Back pain [...] usually feels like an ache

and also this title:

Stomach ache and abdominal pain

It would be particularly interesting to see examples where one of the two words is appropriate when describing physical suffering, and the other isn't.