There are two things the abyssal demon is weak to. The first is demon bane weapons. These are weapons like the Silverlight, the Darklight, and of course the Arclight. Weapon-wise the Arclight has the highest damage per second of any melee weapon. They do consume the pretty rare ancient shards that are more useful when fighting Skotizo or the Zamorak boss in God Wars Dungeon.  The second thing they are weak to is magic. They have no magic defense, so you can just fight them in full armor and still hit them accurately with magical spells.