In theory, the only benefit of FF over APS-C is when using very wide apertures.
For example, you might using a FF camera with a 52mm f/1.2 lens, in order to get the very shallow depth of field this gives and/or the very high light-gathering ability. A good 52mm f/1.2 lens is expensive, but quite feasible (they've existed for many decades).
To get the equivalent effect with an APS-C camera, you would need a 35mm f/0.8 lens. As far as I know, no such lens exists (though it's theoretically possible to construct a lens as wide as f/0.5).
In practice, FF may have other advantages, due to engineering considerations. For example (though I don't really know), it might be that electronic noise from the rest of the camera has more effect on noise during pixel readout when the pixels are smaller.