The larger versions of the Cinch have a wider punch base (12 holes) that allow you to pick which holes get punched and which do not. This allows for more creative bindings—and neater binding for books that don't perfectly match the given size settings—because you can disengage a punch if it will put a hole where you don't want it.
Also, the full-sized versions of the Cinch have a wire-closing mechanism that is adjustable for different sizes of binding wires (3/8", 1/2", 5/8", 3/4", 7/8", 1", 1/8", 1/4"); it sets the width of the gap where the wire sits *before* you apply pressure. The binding wire will fit a bit more snug in the press if the space is matched to the wire first, which may give better control over closing the binding properly.
Binding wire size determines how thick your book can be (the larger the wire, the more pages/thicker the book). The Mini Cinch "works best with 5/8" — which means it might not be able close the binding on a 1" or larger wire very efficiently, if at all; it depends what you can fit under the press plate.
Interestingly, the closing mechanism on the Mini Cinch appears to be slightly curved so that it cups the binding wire; this feature is flat on the larger Cinch (at least it is on my Heidi Swapp version.)