Breath is the air you breathe in and out your lungs. That one is easy. Breadth is a synonym of width (hence the expression hair’s breadth, meaning a very short width), but there are two subtle distinctions breadth and width.
First, whereas width is used for the side-to-side extent of things of all sizes, breadth is generally reserved for things whose spans are especially large. For instance, rooms and streets are often described in terms of width, while hurricanes and large geographical features such as plains and deserts are often described in terms of breadth.
Second, while width is usually used for physical objects, breadth is often used for figurative things. For example, epic stories and far-reaching cultural trends are often described as having large breadths. This is only a usage pattern, however, and not a rule.
Examples
The depth and breadth of Roman history and its influence in Western culture can be a daunting subject to embark upon … [Bullard News]
While tossing in some very classical and very contemporary, too, Wednesday’s program celebrated the breadth of American dance. [Chicago Tribune]
Green Bay Packers quarterback Aaron Rodgers admitted he was holding his breath when the team kicked off late in the fourth quarter. [CBS News]
The whole breadth of society is represented on the various honours lists. [Guardian]
A beautician warned for her bad attitude and bad breath has been awarded more than $8000 after making a personal grievance claim against her New Zealand employer. [Sydney Morning Herald]