This gets argued as a preference question and it is really a mechanics question. The two tools move product in different ways, and on a surface with lines and texture that difference in mechanism produces a visible difference in result.
A brush spreads. Bristles carry product across the surface, which means across the lines as well. Where a line runs perpendicular to the stroke the brush bridges product over the gap and leaves a bead of it sitting in the groove. That is the exact mechanism behind foundation looking fine at 9am and sitting in your smile lines by 11. A brush is excellent at distributing product evenly over a smooth plane, and a mature cheek is not a smooth plane.
A damp sponge presses. You push product into the surface rather than across it, the dampness thins the layer as it goes, and the sponge absorbs some of the excess rather than depositing all of it. On texture that produces a thinner, more even film with less product sitting in the creases. It is also less precise, which is the trade-off.
Then there is the handling question, which usually decides it. A sponge is held in the palm with no grip strength needed. A standard makeup brush is a thin tapered stick requiring a sustained pinch between thumb and forefinger — one of the first grips to become painful with arthritis in the hands, and one of the first to become unsteady with a tremor. A brush with a wide flat handle resting against the fingers is a different proposition entirely, which is why handle shape matters more than bristle quality for this audience.
So the honest answer is both, with different jobs. Sponge for the base layer and the finish, because it controls film thickness. Brush for placing coverage on one specific thing — an age spot, a patch of redness — because it can put product in a small area without smearing it across the whole cheek. The arthritis guide covers the grip side in more detail, and the foundation brush page covers brush choice specifically.