In my opinion, initially at least,
- wide had to do with the space between two objects, how wide apart they are, whereas
- broad had to do with the space taken up by an object, from side to side.
When you say "eyes wide open", you mean there is a great space between the upper and lower eyelids, not that the eyes themselves are particularly large or protruding.
When you say "a broad smile", you do not mean there is a great space between the upper and lower lips, but that the corners of the mouth stretch so far out that the smile takes up quite some space on the face, extends from one side of it to the other.
This distinction seems lost in breadth and width, although I am under the impression that you use breadth when the object has one main, obvious dimension, e.g. "broad shoulders", and width when the object — a table or a board, for instance — obviously has different dimensions: length, width, height, thickness, etc.