The distinction is subtle, for sure. I think piled up is broadly similar to accumulated: the thing is forming a stack. I think piled on is broadly similar to burdened: the thing is being added to a stack that you are carrying. I think you are also right that there is a difference in intensity: things that pile up are more gradual than things that are piled on.
Your example, The work was piling up, makes me imagine a literal stack of documents on your desk steadily growing taller. It is unclear whether there is more work than usual, or if you are working more slowly than usual. You could say the work was being piled on to make clear that the amount of work had quickly increased and I would assume you felt challenged or overwhelmed by the amount of work assigned to you.
In US English, piling on is often used to describe when a group of people collectively oppose another person. It implies that the opposition is unfair or too extreme. Imagine if I suggested a certain restaurant for dinner, then you said that my suggestion was bad, then your friend insulted me for my suggestion. In that case, your friend would be piling on. It evokes the image of a group of people literally pushing you to the ground and jumping on top of you.
I should also mention that there is a idiomatic use of pile up in US English that refers to a collision involving many cars.