apt-get download
will download the package, but not its dependencies, to the current directory.
Use if you want to download a .deb to then tinker with it (uncompress it, poke at it with dpkg, whatever).
apt-get -d install
will download the given package and all missing dependencies to the system packages directory (/var/cache/apt/archives).
Use the second if you want to "pre-download" a set of packages for later installation.
(Derived from a comment on askubuntu: https://askubuntu.com/questions/463380/difference-between-apt-get-d-install-apt-get-download/463391#463391)