Diablo – Does the monster’s class/type matter

Any character (including wizards and witch doctors) can wield any non class-specific weapon, with a few small exceptions.

The class-specific weapons are:

• Ceremonial Knife (Witch Doctor)
• Hand Crossbow (Demon Hunter)
• Fist Weapon (Monk)
• Daibo (Monk)
• Mighty Weapon (Barbarian)
• Wand (Wizard)

There are also off-hand class-specific items, as well as class-specific armor.

In addition to the class weapons, a few classes are restricted from using certain general items. The restrictions are as follows:

• Wizard: No polearms or two-handed maces (other two-handed weapons are OK)
• Barbarian: No staves, no ranged
• Demon Hunter: No two-handed weapons other than bows and crossbows, some skills restricted to ranged weapons.
• Monk: No ranged
• Witch Doctor: No special restrictions

The idea behind having wands for Wizards is that the magic enchantments on them are particularly beneficial to the Wizard so it's more likely you'll be using a wand if you care about general Wizard stats or skills. You are otherwise, however, free to use whichever general weapon you want.

Weapons have become accessories in Diablo 3. Since the game is designed to prefer magic skill attacks over melee, the weapons work like your pants or hat—they add to your stats and attributes, though you may never swing your weapon as a melee attack. You can set melee to a hotkey if you want, though. In your example, yes, the bow and dagger are equivalent. Elemental damage modifiers are added to normal damage. It's all about raising DPS unless you desire an attribute modifier.