Airsoft is a combat simulation where you use harmless versions of proper weapons, sights and all.
Paintball is a game/sport where you run around firing at people with markers that have no sights, ludicrous amounts of ammo, and astounding projectile drop; normally in some sort of arena with large pieces of artificial cover. You're basically playing tag with artificially lengthened arms.
It depends on what you want, really. If you want to play around with your friends and have a hectic but fun experience, go for paintball.
If you want to test your aptitude in a real combat situation: i.e. your ability to take cover, stay undetected, aim and hit, all the while using realistic replicas of real guns, choose airsoft.
Airsoft also contains the gentleman's element where participants have to acknowledge when they've been hit. As such it isn't as fitting for douchebags.
The difference is basically this: Airsoft = Battlefield 2, Paintball = Cod MW: 2/3.
I prefer Airsoft.
It's a game for adults, unlike paintball.
Case in point:
Dr. Pepper Unlimited said:
Paintball. I've never understood, nor will ever understand the appeal of airsoft. I've played it, but it is just too easy to cheat. I'll stick to paintball any time.
People who would cheat if given the chance emigrate to paintball.