unarmed strike is a melee weapon, what makes you think that it isn't. It's not a manufactured weapon, but it's a weapon, and punching is melee.
for the weapon table B stands for Bludgeoning, S is slashing and P is piercing.
To flesh it out, the type of damage you deal matters for some enemies. Like skeletons that takes less damage from any other weapons that doesn't deal bludgeoning attacks. Zombies on the other hand takes less from anything that doesn't deal slashing.
EDIT: Please don't remove your opening posts on your threads. Now others finding this one won't have any context to our answers.