Like Valguarnera, it's a tradeoff between how fast you can move the weapon, and how much mass you can handle. With baseball bats, there has been an evolution and they've reached a balance. With weapons, there are more things to consider.

When you consider how dangerous a weapon is, you also have the size of the wound. It's all about moving / slicing / bruising / squashing the most flesh. With armor considered, you really do need a large and heavy weapon to inflict killing blows. An arrow has high mass, and high speed.. But it doesn't have much stopping power because the wound it inflicts is small.

The rapier is a hefty sword, that can be wielded with dexterity, and can be thrust through armor, and makes a large wound (the blade whole blade is tapered.. the farther you push in, the larger + deeper + deadlier the wound). There's a good chance you're gonna kill the guy with the right stab. Lots of training, and practice will let you know how to get that fatal blow.

Axes weigh an ASSLOAD. Any idiot can get out there and swing an axe. It's nearly impossible to avoid a blow from an axe. The weight + speed means that the axe is going to cut a vicious chunk out of you. VERY high chance of a fatal blow.

No, You do not need a heavy weapon to inflict damage. Heavy weapons infict more damage then light weapons.

In the end we've got many variables
Speed
Mass
Area of Collision
Type of Collision
Size of Wound
Density of Material (you hitting bone? muscle? fat? platemail?)
+tons of misc considerations (range, reach, ease of use etc etc)

And one goal= a dead guy

What do they tell you in algebra? You need a 1:1 ration of equations to variables to solve it =). Hence, many weapons!