Since they both indicate some form of measurement, let us first contrast both words with length. So imagine a square. In this case, width, breadth and length are all equal and are pretty interchangeable (i.e. synonymous).
Now imagine a rectangle. Typically length is used to describe the "longest" side and the "shorter side" can be described as the breadth as in in "the length and breadth of …". So where does that leave width? Well if I had to say "the breadth and width of …" then I would assume breadth still stands for the shorter side and width for the longer side. However, if someone were to say "the length and width of " then of course width then becomes synonymous to breadth.
Now in everyday use… imagine you are standing next to and facing a road (waiting for, say, a bus). The "length" of the road is the distance measured along its curb, and would in all likelihood, be measured in kilometers. The distance from one curb to the other, this distance you cover crossing the road, is typically referred to as the width of the road and rarely as the breadth of the road — unless you are standing in the middle of the road with your arms outstretched towards the curbs… you could poetically say the breadth of the road.
So when to use breadth in common everyday language… well imagine a cube instead of a square. One can easily say the box is x centimeters wide (width), y centimeters long (length) and z centimeters high (height). Try to use "broad" to mean "breadth" in that sentence and you will sound archaic. But then again most people in measuring a cuboid would say width, height and breadth, eschewing length altogether.
So go on. Enjoy the quirks of the English language.