So all distinctions are somewhat arbitrary, since there are always "borderline cases" that have some properties of both, but stars do offer us some rather important differences worth keeping track of that relate to your question. The most important distinction is the one between "high-mass stars" and "low-mass stars," the Sun being in the latter camp (but not too far from those borderline cases). At what mass the line is drawn is not precisely known, and depends a bit on what aspect of stars is being used to make the high-mass / low-mass dichotomy, but that dichotomy is always of crucial conceptual importance. In regard to your question, supergiants are late evolutionary stages of high-mass stars, and red giants are late evolutionary stages of low-mass stars.
The reason this distinction is so important is not that there is such a big size difference between the classes (the variation in size within the classes is even larger), it is because there are so many other important distinctions between the physics going on in the core of high-mass stars versus low-mass stars, and all these distinctions tend to refer to generally the same dichotomy (though not exactly, that's the "borderline" problem). These other distinctions include:
the core of a high-mass star:
has H fusion by CNO cycle, is convective, goes relativistic and vulnerable to contraction before it becomes quantum mechanical. can fuse all elements up to iron, ultimately collapses and can cause a core-collapse supernova
the core of a low-mass star:
has H fusion by p-p chain, is not convective, becomes quantum mechanical and reaches a ground state before it goes relativistic, cannot fuse all elements and often won't go past carbon/oxygen, ultimately becomes a white dwarf.
The problem is that the transition between these types can be for masses on the main sequence anywhere between 2 and 8 solar masses, depending on which particular aspect we are talking about, and some uncertainties and details like rotation or magnetic effects. That's why some authors like to identify a third class called "intermediate-mass stars", to account for that whole transition domain.