A shallower depth of field means that a background can be blurred beyond recognition, which can have a pleasing aesthetic and can be used to focus the audience's attention on particular parts of a scene to tell the story better. Full-frame sensors can help achieve a shallower depth-of-field than smaller sensors — mere fractions of an inch in some cases.
While this can sometimes look nice, the resulting image can also look over-blurred. Pulling focus with a moving camera/subject becomes nearly impossible, resulting in that "seeking focus" look. One of the benefits of shooting with an APS-C size sensor, like the Canon 7D, is that the videographer can open up the aperture (to shoot in lower light), but still have a manageable depth-of-field. On the other hand, with a full frame camera, the aperture must be stopped down further to keep the depth of field in check, which means more light will be needed.