What are the best weapon types for each class in Diablo III? is closed as a duplicate of this question, so I'm going to answer that question here. My answer doesn't quite fit this question, but there's no other place to answer it.
Given two weapons in your inventory that are otherwise identical, there is no advantage to picking a class-specific weapon over a general weapon. A class-specific weapon may have stats that are impossible to find on a general weapon, so in some cases, you may prefer it because the class-specific weapon happens to be better.
It's also worth noting that class-specific weapons are often cheaper on the auction house than the more general weapons (assuming the same stats on both). For that reason, I like to equip class-specific weapons and sell the more flexible weapons.
All that does not mean that all weapons are identical for each class. Several classes, in particular, Demon Hunters, Witch Doctors, and Wizards, have channeling spells. A channeling spell uses your resource (Hatred, Mana, or Arcane Power respectively) with each attack. For those spells, slower weapons are better (assuming that you are comparing two weapons with the same damage per second). The fewer attacks per second, the less resource you use. Two-handed weapons are often slower than one-handed weapons, so there are spell-caster builds that favor them.
Demon Hunters have abilities that require ranged weapons to use. If you want to use those abilities, you need to equip one of a bow, a crossbow, or a hand crossbow. Demon Hunters can also use quivers with two-handed weapons like a bow or crossbow as well as with the one-handed hand crossbow. Combining minimal resource use with maximal choice of abilities means that Demon Hunters favor crossbows with a quiver.
Witch Doctors can go two ways. First, they can choose a fast one-handed weapon like a ceremonial knife. Second, they can choose a slow two-handed weapon. Overall, Witch Doctors tend to favor the one-handed ceremonial knife with a mojo.
A Wizard also has an alternative build called the Critical Mass/Windup Wizard. This build relies on getting as many critical hits as possible. For that reason, you want to maximize attack speed (as well as critical hit chance). The dagger is the fastest weapon that a Wizard can use, so it is optimal for that build. Wands are slower and less suited to that build. However, if attack speed is already high enough, Wizards often choose to use a wand to gain class-specific abilities like Arcane power on critical hits.
A Barbarian has a number of options. They can use a two-handed weapon, dual-wield, or use a shield. In hardcore, shields with a one-handed weapon are most common (defense first). In softcore, Barbarians favor dual-wielding. Barbarians tend to prefer faster weapons which generate Fury quickly to keep their Wrath of the Berserker running.
Monks favor dual-wielding, except in hardcore, where they favor shields with a one-handed weapon. Monks favor fast weapons that generate Spirit quickly. Fist weapons are popular but not overwhelmingly so.
As a rule of thumb, when dual-wielding, the weapon that does more damage per attack should be in the main hand. There are some skills that use just the damage of the main hand weapon.