How do different types of weapons compare? For example, is sword better than a mace, or are they different somehow?
I'll defer to alexanderpas and his excellent answer on weapon materials, but in terms of actual weapon shapes, there are a number of things of interest he hasn't mentioned.
Each weapon has a size and a number of attacks(stab, slash, slap with flat, hit with pommel, etc). Each attack has a contact area, a damage type, a velocity multiplier, and a penetration value.
The larger the weapon the larger the actual contact area and penetration etc(they scale with weapon size, and warhammers and battle axes are bigger than maces and knives, etc). The precise values are in the raws(item_weapon.txt).
Contact area is more contact. Picks have very little, axes have a lot. The more contact area the larger the wounds you inflict, but the more your force is spread out. Axes are good at lopping off limbs because of their large contact area. Spears are not, because they have very small contact area.
Penetration is more penetration. More is always better(does not apply for blunt weapons), but it's usually inversely proportional to contact area. Axes rarely stab people in the heart, but spears and swords do.
Velocity multiplier is basically an effect of leverage and long handles. The velocity multiplier multiplies the impact velocity. This is part of why whips are so deadly(velocity multiplier 5000, compare axes 1250 at most).
The whole thing is a bit of a balancing act, especially considering the variety of foes your militia will need to handle. Swords are good all-rounders, with stabbing and slashing and slapping, but don't excel at anything in particular. Warhammers and maces are for armoured targets and hard-material forgotten beasts. Edged weapons will have a hard time scratching an amethyst scorpion, but a silver mace can chip at it because a blunt weapon doesn't have to be harder than the target to deal damage. Spears are good against very large targets like buffalo or fleshy forgotten beasts, where blunt weapons are just absorbed into the layers of flesh or skin and edged weapons can't lop off limbs because the limbs are too fat. Spears can penetrate deeper into the fleshy bits and injure organs and muscle layers, which bleed more than fat or skin. Axes are great for large numbers of small foes, as it can incapacitate them quickly before they overwhelm the wielder.
For armored or similarly rigid targets, use warhammers. Big fleshy things? Spears. Loads of (relatively)little things or perhaps something with a poison sting? Axes to chop bits off before they bite/sting/dogpile you. Need all-around performance? Swords.
The edge cases have been softened a bit since pulping was introduced (it's now possible, if tedious, to kill cave blobs with maces, for instance) so a repeated beating with any weapon will eventually kill nearly anything. If you're trying to hammer something with no skeleton be prepared for a lengthy evening and perhaps a quick break for refreshments and a nap.
Warhammers and Axes beat the pants offa Maces and Swords in their respective damage categories (i.e., edged vs. piercing)
Spears are off in their own little group.
Generally speaking, Axes do deadly deadly things to unarmored critters, Warhammers quickly cause crippling wounds (leaving enemies free to be maimed by the aforementioned axedwarves) and spears are useful against larger enemies such as megabeasts, forgotten beasts, and titans.
This thread might have the more detailed information you're interested in:
There are two different types of attack, namely blunt and edged (sharp); In addition, each different weapon and material has different stats for Yield, Fracture and Elasticity, which are used to simulate proper Stress & Strain.
As this makes it impossible to give specifics, the general consensus is:
Notes: Copper is better at piercing Iron armor than Bronze is, but Bronze pierces Copper and Bronze armor and better than Copper does
Notes: All six non-adamantine metals perform nearly identically. Steel has a slightly higher rate of critical wounds, while silver is slightly more likely to penetrate armor.
This came up in another question, but there is a long chart here:
I'd summarize it, but it boils down to: there aren't really any hard and fast rules on the issue.