vi on your computer is almost certainly a symlink to
vim, renamed executable, or something similar rather than an actual
vi executable. The Vim package(s) in the repositories are built with different switches (i.e. features).
Small note first: there is no program called Vi/Vim. Vi and Vim are two separate programs. Vi is an editor created in the 1970s. Vim is a clone (not a fork!) of Vi created in the 1990s.
Vi was, until the early 2000s, under some restrictive licensing, not to mention having limited development after the 1980s. Since Unix clones were generally expected to provide a vi executable, open source operating systems such as BSD and Linux turned to Vi clones. In Linux's case Vim is usually selected.
In order to ensure compatibility, a symlink, hardlink, renamed executable, or some other thing is created at /bin/vi (or /usr/bin/vi) that, rather than launching Vi, launches Vim.
As far as the additional package(s) in the repositories, well, Vim has a lot of build switches, controlling things such as whether Python can be used in plugins and which version of Python; whether there's a GUI and if so, which GUI (GTK+, Motif, Athena, Win32, etc.); whether there's clipboard support; whether there's mouse support; and many, many other things. Linux distributions will typically pick a somewhat limited build to include in the base image, and make additional variants available through the repos. And of course, if you're not satisfied by the options there, you can build your own and select exactly which features you want.