Ubuntu – How are packages suggested and what is the difference between extra and new packages

suggests and recommends are not the same thing.

Package A depends on Package B if B absolutely must be installed in order to run A. In some cases, A depends not only on B, but on a version of B. In this case, the version dependency is usually a lower limit, in the sense that A depends on any version of B more recent than some specified version.

Package A recommends Package B, if the package maintainer judges that most users would not want A without also having the functionality provided by B.

Package A suggests Package B if B contains files that are related to (and usually enhance) the functionality of A.

See The Debian GNU/Linux FAQ for details.

By default, apt-get installs recommended packages.

Your option --install-suggests adds in the suggested packages and their suggested dependencies.

--install-suggests
    Consider suggested packages as a dependency for installing.
    Configuration Item: APT::Install-Suggests.

see apt-get(8).

Not much. apt is a new command that supposed to merge several functions from apt-get and apt-cache into one command. It's still a little rough around the edges but here's the command listing from --help:

Basic commands: 
 list - list packages based on package names
 search - search in package descriptions
 show - show package details

 update - update list of available packages

 install - install packages
 remove  - remove packages

 upgrade - upgrade the system by installing/upgrading packages
 full-upgrade - upgrade the system by removing/installing/upgrading packages

 edit-sources - edit the source information file

The equivalent functions are designed to work in similar ways but it's not a proxy command (it's not calling the old ones - it's a new interface directly onto the Apt libraries) so there may be some edge-case changes.

There are also some obvious omissions (download, policy, etc) that power-users will miss and there are a whole raft of undocumented commands (purge still works but I can't find anything on it).

16.04 Update: A lot of the omissions have now been included but aren't yet documented, nor do they have Bash-completions. It's a shame it's taking this long to implement functionality that already exists in the codebase but oh well. My advice is that if you're used to an apt-{get,cache} command, try it on apt. It might work.

There's also a DIFFERENCES TO APT-GET(8) section in the man apt page that's interesting:

   The apt command is meant to be pleasant for end users and does
   not need to be backward compatible like apt-get(8). Therefore
   some options are different:

   ·   The option DPkgPM::Progress-Fancy is enabled.

   ·   The option APT::Color is enabled.

   ·   A new list command is available similar to dpkg --list.

   ·   The option upgrade has --with-new-pkgs enabled by default.

And if you want Bash-completions, I've had an attempt as writing a completions file for it already. These are included with later Ubuntu installs.