There are some damage calculations happening behind the scenes. Here are some notes on the weapons.

The base weapon power of the Fauchard is -1 less than base-game spear and -2 than sword. Additionally, there is a flat -2 damage built into the calculation. This is a spear which requires Sword Skill to improve accuracy. Investing in neither Spear/Sword Skill means this weapon is very weak.
Investment Preview:

• Investment: +3 ONLY Spear
  dmg equal to sword, less than a spear -- 92% acc
  
• Investment: +3 ONLY Sword
  dmg equal to spear, less than a sword -- 98% acc
  
• Investment: +3 BOTH Spear & Sword
  dmg equal to a sword, more damage than a spear, slightly less acc (98% vs 100%), different range: 3 cleave vs adjacent for Sword vs 2 for spear

Weapon tradeoffs: Requires 4 stars in Swords to get 100% accuracy. More range than either a sword or a spear, but requires careful setup to not hit allies with the cleave. It is 1hnd, so it can be used as an offhand when the time is right. Giving a Fauchard to a character who has heavily invested in Swords is better (more Acc, more damage than a spear) than giving it to a character who has invested only in Spears (less Acc).

The base weapon power of the Urumi is -1 less than base-game flail and -2 than sword. Additionally, there is a flat -1 damage built into the calculation. This is a flail which requires Sword Skill to improve accuracy. Investing in neither Flail/Sword Skill means this weapon is very weak.
Investment Preview:

• Investment: +3 ONLY Sword
  more dmg than flail, less than sword, 98% acc
  
• Investment: +3 ONLY Flail
  equal dmg to sword, less than flail -- 92% acc
  
• Investment: +3 BOTH Flail & Sword
  equal in damage to a sword, more damage than a flail, slightly less acc (98% vs 100%), different range: max range 2

Weapon tradeoffs: Requires 4 stars in Swords to get 100% accuracy. It has more range than either a flail or a sword. Giving an Urumi to a character who has heavily invested in Swords is better (more Acc, more damage than a flail) than giving it to a character who has invested only in Flails (less Acc).

While loaded and with no points in Fist Skill, Bolas is -3 damage from a crossbow (-1 weapon power, -2 built in damage reduction). However, it debuffs movement while loaded and it can attack with a powerful punch to reload. Putting points in Fist Skill will bring the weapon up to par with an xbow. Additionally, attacking with it unloaded adds +2 flat damage, making it stronger than an unarmed or gloved attack, regardless of Fist or Aim Skill.
Weapon tradeoffs: No downtime like with an xbow because you can attack to reload. Both the debuff and the +2 damage alternate, so you can't use one two turns in a row. If you use a bolas as an off hand weapon, you lose your offhand slot to equip a glove, which comes with stat bonuses. There's a longer range than just a punch while loaded. This weapon is weaker in power than an xbow (-1), but stronger than gloves, which have no inherent weapon power. Because this alternates between two "separate" weapons (ranged xbow, melee fist), your available skills alternate.

The Vagari has the same tier weapon power as a bow (so less than a base-game sword). The damage is balanced so that at sword range, assuming your sword skill is equal or higher to your bow skill, a base-game sword will out damage a Vagari at melee range. If you pump ONLY bow skill, Vagari will do 25% more damage than a base bow at full range. If you pump both sword and bow more or less equally, it will be weaker than a sword but stronger than a bow.
Weapon tradeoffs: This is a bow, so only ranged skills are available and it requires 2 hands. However, it can attack at melee distance, it gains 25% extra damage at full range, and it can only attack in straight lines.

Gains increasing MP Heal per weapon tier (bone through mythril). MP Heal triggers even when attacking an empty square. (Note: Hover over your character in the attack preview to see your MP Heal)