Mother Nature is resilient and the stump just might send up new shoots trying to rise again. Fungi, mold, termites, and ants are just some of the common things that stumps regularly attract. If you originally had the tree cut down because you discovered it had a disease or had a pest issue, the stump can continue to spread those issues even after the removal of the main trunk.
Some people just want stumps gone because they don’t like the way they look. Other people may be concerned about them being a tripping hazard or damaging their lawn equipment.