I\'m familiarizing myself with the whole homebrew kit and the documentation is rather poor. What is a cask, Cellar and a tap?
I personally differentiate brew & cask just based on 1 of the main (if not the only main) differences, of graphical GUI applications being managed by cask rather than just brew. (There are other differences too which I don't know as much about).
Below adapted from 'Graham Miln' on apple.stackexchange:
brew is the core command for the Homebrew project.
The missing package manager for OS X
Homebrew installs the stuff you need that Apple didn’t.
Homebrew typically deals with command line software (not graphical GUI applications). Most of the software is distributed under an open source licence.
brew cask is an extension to brew that allows management of graphical GUI applications.
Homebrew Cask extends Homebrew and brings its elegance, simplicity, and speed to OS X applications and large binaries alike.
Cask deals with a mixture of software and licences.
Cask offers a way to command line manage the installation of graphical GUI applications.
Availability through brew or cask does not imply any specific licence.