Properties with Belt-driven Chains?
The main difference with belt drive is that you can't break the belt, so you need a specially built frame that allows the belt to be fitted. On a chain-driven bike the chain usually runs though the frame between the seatstay and chainstay. This makes retrofitting a belt drive impossible on most bikes. Similarly, you're limited to the belt lengths available from the manufacturer.

I have seen bikes with exposed belt drives but that does not seem like a good idea to me. Once you get mud or dust in there it's going to grind away the belt and pulleys, and parts are usually expensive (because they're uncommon). I don't know much about how they wear as I've never seen one that's had much use (which in itself should tell you something).

Once the belt is fully enclosed it becomes just like an enclosed chain drive, except that it can't be serviced by bike shops. I suspect this is the real deal-breaker for most people.

Belts usually have higher losses than chain drives, making them unattractive for high-performance bikes and hindering their adoption in other bikes (because much of the profit for bike shops is in the more expensive bikes).