Chains vs. belts vs. shafts
Chains are heavier and noisier than belts in an engine but they rarely, if ever, break. Belts are lighter so they lower rotating mass in the engine and they are also quieter. The only downside to this is the added maintenance of replacing the belt periodically. On some engines, if the belt breaks while the engine is running it could cause catastrophic damage to the valvetrain or even the pistons.
On motorcycles, a chain drive system is much simpler and lighter and also delivers more power to the wheel. Shaft drive is more complex, heavier and of course more expensive. About the only real benefit to shaft drive is that they require less maintenance, a lot less, and they are a bit smoother.
Belt drive is great unless it breaks...then you are screwed. Chains are much stronger than belts.