I have designed chargers.
I couple of people covered the problem correctly.
It generally has NOTHING to do with the fact that it is a 700ma or 1 amp charger. It has everything to do with needing to short the data pins.
First, install the app "battery monitor". Open up battery monitor and plug the phone into your AC charger. Battery monitor will indicate that it is plugged into an AC charger, and after a long while will show a high charge current. Then plug it into a USB port, Battery Monitor will indicate it is a USB port, and will eventually show a lower charge current. Plug it into your car charger, and it will likely show that it is a USB port, not a fast AC charger.
If you are careful, you can put a small piece of foil in the large USB port in the charger, and short just the two center data pins. That should give the high AC charge rate. And just glue the USB plug in to keep it from moving the foil. I typically open the charger and short it out inside.
The most current that I have seen drawn by the SIII is 800ma. So a 700ma will only be very slightly slower, and a 1 amp would be best of course. Any more is a waste. I have seen cables with some really small gauge wire that couldn't do the job. But that is rare.
This is all by design. It is part of the Micro USB standard. When plugged into a PC USB port, you have to limit the current. The SIII and all other phones I have tested, limit it to about 300ma. When plugged into AC (the data lines are short), the phone can safely pull more current. Usually in the 700 to 800ma range. I have yet to find a cheap car charger that had the two center USB data pins shorted as they are supposed to be.