Swath is only a noun. It refers to (1) the width of a scythe stroke, (2) a path made by mowing, or (3) something likened to a path made by mowing. Today, it’s usually used in the third, figurative sense.
Swathe is usually a verb, meaning, primarily, to wrap or bind with or as if with bandage. It also functions as a noun meaning a wrapping or bandage, but it is rarely used this way in today’s English.
Examples
Swath
Doherty represents a large swath of suburban New Jersey where resentment toward the extra funding of low-income schools runs deep. [NPR]
Fresh fruit and vegetables surged 11.3 per cent last month after Cyclone Yasi cut a swath through banana plantations in Queensland. [Herald Sun]
But after swiftly capturing swaths of the country, pro-Ouattara forces have met fierce resistance in Abidjan in the past four days. [Guardian]
Swathe
Rotting drainpipes, embedded in the school’s interior walls and swathed in asbestos, are difficult to reach and repair. [NJ.com]
Dr Christina Brunner then ordered nurses to swathe the unconscious woman in bandages and not report the accident … [Mirror]
Wounded Warrior was held together with a sturdy swathing of duct tape. [Colorado Springs Gazette]