Like many of our shortest English words, sit and set have lengthy entries in the Oxford English Dictionary. Some of the definitions overlap. Some are interchangeable.
The most common uses of sit and set are similar to those of lay and lie. “To sit” is to be seated. “To set” is to place something somewhere. In these contexts, sit is intransitive and set takes an object.
Mixing up sit and set is not as common as mixing up lay and lie because the principal parts of sit and set are completely different:
sit, sat, (have) sat, sitting
set, set, (have) set, setting
However, because sit and set have so many additional uses, efforts to state a hard and fast rule as to when to use one and when the other are futile. That fact doesn’t stop people from trying. I read a comment asserting that “animate objects sit, whereas inanimate objects set, and that’s that!”
If “that were that,” the following statements would represent standard usage, but they don’t.
The flowers were setting on the table and the men’s tuxes were draped over chairs.
We were surprised by the beautiful gift-wrapped package setting on our bed.
Both “flowers” and “package” are inanimate objects, but sitting is the verb called for in both statements.
The meanings of sit listed in the OED include this one:
a. Of things: To have place or location; to be situated. Ex. There were a dozen eggs still sitting on the front porch and the dustbin sat at the back of the house where the binmen had left it.
The flowers were sitting on the table and the package was sitting on the bed.
The expressions “to sit well” and “to set well” have differing meanings.
A certain plan may not sit well with voters. Here, “to sit well” means something like “to please” or “be agreeable to.”
A jacket may be said to set well on the shoulders. The OED definition for this sense of to set is,
To have a certain set or hang; to sit (well or ill, tightly or loosely, etc.).
In texts written about clothing, you will also see “to sit well” used in the same sense:
Just because you can squeeze yourself into a garment doesn’t mean it sits well.
Trousers with a wider waistband sit well.
When speaking of clothing, “to set well” and “to sit well” seem to be interchangeable.
In the matter of liking or not liking legislation, “to sit well” or “not to sit well” is the way to go.
In speaking of an object that has been placed somewhere, the choice is “sitting.”