Nowadays, these three terms – barrel, cask or keg are considered to be synonyms.
Historically, each of these words had different meanings, depending on the regions and countries. However, the size specificity has disappeared through the last centuries.
In French, there are also different terms, such as tonneau, fût or barrique. In Burgundy, we even use another word : the pièce, whereas in Bordeaux, winemakers talk about their barriques, and there is actually a 3 liter difference between these two kinds of barrels!
For more information, see our FAQ dedicated to the different barrel sizes.