Java's
/usr/bin/apt is some "Annotation Processing Tool", I don't know anything else about it, but it's obviously not an update/upgrade utility and it's not related to Debian's Advanced Package Tool. Guess it might conflict with
/usr/bin/apt from APT 1.0 - if the Oracle guys didn't rename it later, of course.
As for Mint's
/usr/local/bin/apt, it should work the same way in Maya and Qiana.
Also it always gets chosen first - as I said before,
Since /usr/local/bin usually comes first in the $PATH, when you simply run "apt" in the terminal, Mint's script will be executed.