Shallower depth of field
Among other differences, with a full frame sensor, you can get more shallow depth of field for a given focal length and aperture. With focal length and aperture held constant, the closer the camera is to the subject, the shallower the depth of field; the further, the deeper.
With a full frame sensor, you have to move closer to the subject to fill the frame than you would a crop sensor. Thus, less depth of field. This is an advantage for many types of photography, such as portrait, where you want a background blur. When people, accustomed to digital point-and-shoot cameras, see portraits taken with a DSLR, one of the first things they notice is the background blur (called bokeh), since it's just about impossible to achieve that effect with the tiny sensor on a P&S.
Of course, an APS-C and APS-H sensor can also provide very pleasing bokeh, but the affect is more profound as the sensor size increases.