For those who aren't familiar with camera terminology: 'full-frame' refers to the size of the image sensor. A full-frame sensor is a full 36mm x 24mm, equivalent to traditional 35mm film. Most DSLRs (as well as Sony's NEX series cameras) use smaller APS-C sensors which are 25mm x 17mm (varies depending on manufacturer).
Larger sensors allow for larger pixels, which in turn capture more light. This lowers the noise and improves low-light performance significantly. Furthermore, larger sensors have a more shallow depth of field for the same aperture size, making it easier to provide out-of-focus backgrounds for more artistic shots.