1) Virtual instruments. The touch-screen piano is a cheesy novelty without touch response. For it to be truly useful, of course, pre-hoc feedback will be necessary, but pressure sensitivity will be one component that advances the touch screen instrument.
2) Hover effects. Previously afforded only to devices with a mouse and cursor, the hover effect is essential to tooltips, link-testing, and a sort of visual substitute for tactility, that is an electronic equivalent of running your fingers over something to probe and test it.
3) Replacing double clicks or push-holds. The double click is mechanically incompatible with a touchscreen. A double-tap on a touchscreen requires a user to physically lift their finger off the screen and re-engage it; it's simply too slow to work. Meanwhile a push-hold where a user holds their finger on a control for slightly longer inevitably feels wrong. Even if the requisite delay is well calibrated, such as on BlackBerry X, it is an intrinsically unsettling feeling. Rather than making it a difference of time, then, pressure-sensitive screens can allow a difference of force. A tap can activate a lighter function and a press can activate something more consequential.
Specific applications include dragging objects, renaming or editing, and popping up context menus.