Does a ACE wrap for an ankle sprain actually help? It doesn't seem to change the swelling much. What's bad about swelling? Does swelling cause damage?

Yes. An ace wrap can help reduce dependent swelling after an injury. You must also rest, and ice and elevate the foot above heart and head x 30 minutes 4 x/day, until the swelling is gone. If swelling persists, no damage is done, except that you will probably continue to hurt and likely will get stiff in the ankle. Controlling swelling allows for quicker, pain free recovery.
Yes it can help. With any acute injury, such as an ankle sprain, it is important to reduce swelling an limit movement of the ankle. Follow the rice principle: rest, ice, compression and elevation. Excessive swelling can delay the healing process and cause continued discomfort. See your podiatrist if dicomfort persists or progress is slow. Dr l.
Ankle Sprain. When applied appropriately, an ace wrap will help push the swelling about of the area. Swelling is a natural response to an ankle injury and it's main issue is that it causes discomfort and isn't helpful in healing because it can actually inhibit the inflow of healing factors in the blood stream by closing down smaller vessels.