Breath: the process of taking in air to get oxygen into the lungs; the air taken in or pushed out during breathing; small amount of air or wind
Taking a deep breath, she gave her manuscript to the printer

Breathe: the act of taking air into or out of the lungs. Also refers to letting air through a material (e.g. letting red wine breathe or choosing a fabric that can breathe)
It can be harder to breathe at high altitudes.