The Pickaxe is a tool used for gathering resources. Although you can use it to deal a very slight amount of damage, it is not technically a weapon because it cannot be crafted, replaced, or damaged. The pickaxe can be evolved to be stronger in the Skill tree.
In Battle Royale, the pickaxe functions in almost the same way as it does in Save the World. You are able to gather resources with it, and it can function as a last resort weapon that deals 10 damage if need be.
One difference that the Battle Royale pickaxe has from Save The World is that you can purchase cosmetic skins for it, making it look and sound different from the default pickaxe. These cosmetic skins offer no advantage in gameplay.