What's the difference between `brew tap caskroom/cask` and `brew install brew-cask`?

Homebrew is written in Ruby, which uses Perl's motto of "There's more than one way to do it" to cater to every special interest, to the confusion of newbies.

bmike thinks this is important:

Also - the handling of taps / casks has changed over time. On Homebrew 1.2.4 you can jump straight to installing from cask and the best guide on getting started that's current is:

  • https://github.com/caskroom/homebrew-cask/blob/master/USAGE.md

But it confuses me as brew install caskroom/cask/foo does the same as brew cask install foo.