The Julia installation that you get if you go to https://julialang.org/downloads/ and download Julia 1.6.3 dmg for Mac is exactly the same as the installation that Homebrew's Cask currently gets, which is evidenced here: https://github.com/Homebrew/homebrew-cask/blob/master/Casks/julia.rb
Julia behaves differently once you've actually installed the Homebrew Julia 1.6.3 Cask versus the Julia 1.6.3 from https://julialang.org/downloads/. I first experienced this when trying to use PackageCompiler.jl to make a shippable application using Homebrew's Julia 1.6.3 Cask install. With this install, the app continued to fail despite all my efforts. When I switched to the Julia 1.6.3 dmg install from https://julialang.org/downloads/, the PackageCompiler app worked and was relocatable.
So just a heads up, these two methods install Julia differently on the system, resulting in Julia behaving differently in some cases.