To chafe is to irritate by rubbing. For example, a poorly made shoe might chafe your ankle. The word is often metaphorical; for instance, you might be chafed by a bothersome coworker who talks too much. Also, chafe sometimes becomes intransitive, taking the preposition at (or sometimes under), so that we might say you chafe at the coworker’s bothersome behavior (instead of saying you are chafed by it).
Chaff is primarily a noun. It refers to (1) husks of corn or grain separated by threshing, (2) straw or hay fed to cattle, and (3) worthless or leftover matter. The word is often embedded in the figurative expression to separate the wheat from the chaff, which means to separate the superior from the inferior.
At present, the 49 per cent Turkish government-owned airline is chafing at the limitations of Ataturk airport, its current hub. [Financial Times]
When the young man would carry the weight in a backpack, his shoulders would become chafed. [The Promises of God, Clarence E. Warner]
In Bahrain, the Sunni monarchy rules over a Shiite majority population that has begun to chafe at the lack of political and basic human rights there. [Christian Science Monitor]
It’s no secret that Milwaukee is stacked in the “folk rock” department – which can make separating wheat from chaff a daunting task. [Milwaukee Magazine]
To process seeds for storage and replanting later, you must first remove the chaff; otherwise it will rot. [Food Not Lawns, Heather C. Flores, Toby Hemenway]
The liver chestnut Irish Hunter (Irish Draught-thoroughbred cross) enjoyed four meals a day, including sugar beet and chaff, to maintain his condition. [Horsetalk]