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).
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I've never ridden a belt-drive bicycle, but I've ridden both belt and chain drive motorcycles and the belt drive is considerably smoother and quieter.
One of the features being used to tout belt-driven bikes is that they don't require lubing, so that would be a plus.
In addition to not needing to take the time to properly lube your drive-train, the lack of lube would mean less attractant for dirt/dust etc.
Each manufacturer would probably have their own recommendations about operating temperature range and how often to change them.
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Belt drives can't be used with derailleurs so the bike will be single speed or need an (expensive) hub gear.
There were some cars that used a belt drive, usually with a v-belt and split pulleys to give a continuously variable gear -
haven't heard of any bike doing this.
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What are other properties with belt-drive chain?
Main differences compared to using a chain: belt does not need lubrication (less maintenance) as it is not lubricated, it attracts less dirt, and is cleaner, even if it is exposed (less danger to mess up clothing)
it has more friction/power loss than a chain (but less than a badly-oiled chain)
it stretches more than a chain under heavy load (so less suitable for high-performance cycling)
it needs a frame that can be split, because the belt cannot be split it cannot be combined with a derailleur Where is the idea got?
The idea is very old.
It was used in the first steam engines (and probable even earlier).
Are belt-driven chains used in cars or some machines?
Yes, they are quite common.
In particular, the alternator in a car is usually driven by a belt.
Are they durable?
Do they last in all kind of weather?
Like in freezing winter to sunny summer?
I don't know specific numbers, but they should last at least as long as a chain, and modern belts should have no trouble with high/low temperatures (they are used in cars as well, after all).
Do they stretch like steel chains?
Actually, steel chains don't strecht (noticeably), the just get longer from wear.
Belts on the other hand do strech under load (see above).
When should you change them?
No idea, but they should last considerable longer than a chain (less wear).
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(I know that this is an old question, but nobody has answered it with real-world experience.)
I have a 'Scott Venture 10' as a suburban utility bike since two years.
I don't ride it as much as my road bike, but it has seen its share of use.
It has a carbon-fibre reinforced drive belt, and Shimano Alfine hub gears.
There a couple of advantages that I can think of:
It doesn't require any maintenance.
(Or at least much less than a conventional chain.
Especially when riding in rain.)
I don't have to tuck my trousers into my socks to keep them clean/whole.
When walking past the bike in the shed I don't risk ruining my trousers by rubbing against a dirty chain.
According to the manufacturers the estimated life of a belt is significantly longer than that of a chain.
The main disadvantage is (as mentioned above) that it requires a split frame, which one would naively think would reduce rigidity.
The manufacturers claim that this is not the case, however...
It's also quite a bit more expensive (and specialized) to replace.
An ideal chain is more efficient than a belt, but in most real-world scenarios there is no real difference.
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This reply about belt-driven belt got my interest  here . It does not need lubrication at all according to the replier. What are other properties with belt-drive chain? Where does the idea come from? Are belt-driven chains used in cars or some machines? Are they durable? Do they last in all kind of weather? Like in freezing winter to sunny summer? Do they stretch like steel chains? When should you change them?
Properties with Belt-driven Chains?
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<chain><belt-drive>
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Belt drive is not exclusive to bikes. It dates back to the first steam engines, and is also used in car alternators. One of the main problems is that belt-driven bikes have higher losses in terms of friction and power, and stretch more under weight, which compromises performance. Moreover, the structure of belt-driven bikes is more specialized, hence they are more difficult and expensive to replace. There are also limitations regarding the length of the belt and the gears. On the other side, the advantages of belt drive include less maintenance, as lubing is not required, greater product life and a cleaner bike. Regardless, some argue that, in real terms, there are no great differences between chain and belt drive. One of the main problems is that belt-driven bikes have higher losses in terms of friction and power, and stretch more under weight, which compromises performance. 
One of the main problems is that belt-driven bikes have higher losses in terms of friction and power, and stretch more under weight, which compromises performance.  You might want to consider that the structure of belt-driven bikes is more specialized, hence they are more difficult and expensive to replace. There are also limitations regarding the length of the belt and the gears. The advantages of belt drive seem to include less maintenance, as lubing is not enquired, greater product life and a cleaner bike. Belt drive is not exclusive to bikes. It dates back to the first steam engines, and is also used in car alternators. In real terms, there are no great differences between chain and belt drive.