Wide makes reference to an extend of space something occupies but not necessarily fills.
What is wide? An entrance, a path, a road, a passage, an aperture, a door, eyes wide open...
What is broad is wide as far as its limits go (wide refers to limits), but at the same time something broad the space between the limits.
A broad plank, a broad border, a man with broad shoulders, broad horizons, a broad expanse of land...
A wide difference, but a broad line of distinction.
Wide is opposed to close, broad is opposed to narrow.