Are class-specific weapons better than other weapon?
Solution 1:
Class specific weapons can provide some class only enchantments that you cannot get on standard weapons. Whether or not this is worth a trade off for damage, that is entirely up to you to decide.
For instance, you can find a demon hunter only 1 handed crossbow that provides hatred regeneration or a wand for mages that provides improved arcane pool.
Like James mentioned in his answer, the demon hunter is the only class that has a unique situation where you MUST have a ranged weapon equipped or you can't use his ranged attack skills. And his one handed crossbow is the only ranged weapon you can dual wield.
Solution 2:
From my own experience (read below), as long as you can use the weapon and it has better stats, it will increase the effectiveness of your characters abilities which are all weapon damage based.
However, as I approached level 10 or so I started finding versions of weapons that could only be used by their respective classes. This was most notable with bows/crossbows that could only be wielded by a demon hunter. Also some 2h weapons that I believe were either for the barbarian (Hammer) or monk (Daibo) sort of deal.
I cannot say for sure for another couple of weeks as they had us pretty limited in the open beta this weekend, but I do look forward to getting more information in this area. I played up to level 7 using my noob wand on a Wizard until it was pointed out that I can use just about any weapon and my skills will still work. NOTE: This did not work for the Demon Hunter actually. Using a melee weapon stopped most 'shooting' base skills from working.
Solution 3:
Class-specific items do not have any intrinsic benefits, other than the ability to spawn with class-specific affixes, as yx mentions.
On a pure stats basis, I do not believe class-specific items are any better either. In every category I checked, class-specific weapons had equal or lower DPS than the class-neutral variant, and the same number of possible magic affixes.
Here are some data points:
2-handed Monk weapons:
Daibo: http://us.battle.net/d3/en/item/daibo/
normal: 535.2 dps
crafted: 442.2 dps, 4 magic properties
Staves: http://us.battle.net/d3/en/item/staff/
normal: 584.5 dps
crafted: 484 dps, 4 magic properties
Polearms: http://us.battle.net/d3/en/item/polearm/
normal: 601.3 dps
crafted: (none)
1-handed Monk weapons:
Fist weapons: http://us.battle.net/d3/en/item/fist-weapon/
normal: 248.5 dps
crafted: 210.0 dps, 6 magic properties
One Handed Swords: http://us.battle.net/d3/en/item/sword-1h/
normal: 248.5 dps
crafted: 210.0 dps, 6 magic properties
One Handed Axes: http://us.battle.net/d3/en/item/axe-1h/
normal: 285.3 dps
crafted: 238.5 dps, 6 magic properties
Of course this does not affect set or legendary items. But what it comes down to is, use the item with the best stats and bonuses present (as long as you like its attack speed), regardless of item type.