I don't entirely agree with the accepted answer.
I have read somewhere that the extra energy required to start the engine is about a second or so of the amount of fuel it would consume when continuously running. So, it is definitely clear that the start/stop system saves energy. If starting an engine required a lot of energy, where would the extra energy go, then? It wouldn't make sense knowing the laws of physics. The rotational kinetic energy of the engine is anyway so low that it stops in less than a second if fuel supply is cut. The efficiency of starter motor (and the alternator) might not be 100%, but even a 10% efficiency should mean that starting takes about the same energy than turning the engine for 1 second.
However, does it save enough energy for it to pay back? Probably not. I have 60 000 km on my 2011 Toyota Yaris, and it has a functionality in the car computer for telling how long the engine has been off. If I recall the figure correctly, it has been off for about 13 hours. If the car lifetime is 300 000 km, it will be off for about 65 hours during the lifetime of the car. How much fuel does it save, then? Assuming that 0.7 liters per hour are consumed by an idling engine, it is 45.5 liters of fuel saved. In Finland, this costs a bit more than 60 EUR (in the United States, it would be much, much cheaper due to the lower taxation level).
Now, what does the start/stop system require, then? Firstly, it requires a heavierweight battery. The start/stop system battery in my car is not an AGM battery but it is a flooded battery where the plates are thicker. Probably a bit more expensive than a regular flooded battery. Secondly, it requires a more expensive starter motor that can withstand more start cycles. They at Toyota have indeed taken this into account, and the starter motor has a certain amount of start/stop cycles it can withstand, and it requires replacing if the amount of cycles it can withstand is exceeded (which probably will never happen for most users). Thirdly, it requires the computer program to control the starter motor, but this is merely constant R&D cost without variable per-car cost.
What are the disadvantages, then? I assume that premature battery wear might be a problem. I have noticed on my car that the lights have recently started to momentarily dim when the start/stop system is doing its job and re-starting the engine. But my battery has been in use for 4.5 years, and it might in fact be a good lifetime for a battery.
I have recently started holding down the clutch when on a stoplight, because I don't believe the savings are that great, and because I'm concerned about premature battery failure.
Why do car manufacturers install start/stop technology then if it doesn't pay itself back? I assume the reason is that taxation (annual tax and vehicle purchasing tax) in most countries is heavily based on fuel consumption, and because the fuel consumption driving cycles have an excessive amount of time when the car is stopped. Most drivers never stand at stoplight for as long as in the driving cycles.