A FF DSLR: bigger sensor (35mm film equivalent size)
A DSLR: smaller sensor compare to a FF body
My own view is that a DSLR will still make full use of the lens area, not only the centre portion only. Just that because the sensor is smaller, the rest of the images outside the perimeter is not captured or
what we called crop off.
For a FF DSLR, the whole image is captured on the sensor. So any lens of poorer quality the defects of images at the edges will be shown clearly.
It doesn't make full use of the lens area unless the lens is designed for a cropped sensor. A FF frame lens will have an image circle larger then the sensor. The sensor simply makes use of the centre portion of this image circle. Only FF sensors and 35mm film can make use of the entire image circle( well a rectangle that touches the corners of this circle at least). The "rectangle" area used by a cropped sensor is cropped by the same factor as the sensor.