Dpkg is the low level tool for Debian packages.
Apt-get is the original frontend for dpkg. It is a full featured tool that lets the user give commands to dpkg, along with apt-cache, which displays information to the user.
Apt is a high level tool for user friendliness. It combines some features from apt-get and apt-cache, as well as adds progress bars and other quality of life features. It also strips down some features the average user doesn’t use.
So neither is a wrapper for the other. They are two similar tools that do the same job. Apt-get is better for scripting due to being a more rigid tool while apt is nicer for end users.
When I started on Debian, there was only apt-get. (And dpkg if you manually pulled .debs from somewhere).
Then a little while later, there was aptitude, which was nice.
apt the command didn’t show up until 2014.
