We people have issues like that too. I am not so sure about Samsung S3, but I am sure about Android, I had an android phone, whose Drivers were perfect! I have checked for Drivers by the CD that came.
The issue is either not because of the cable, because if it charges well then it must connect to PC too, then where is the issue?
Here are the two major reasons:
Please check that your USB supports fast data transfer (USB 2.0), because Android would try to connect to a faster one.
Please note that there is alot of issues with Android itself too. Like many other apps running. When you connect to the USB, you get a popup that connecting usb storage would stop many other apps bla bla. So that is a reason too as android does not want to stop some personal apps from working. For this, you can restart the phone. This way this issue would be resolved, I have checked this one more than 5 times.
And USB tethering is just related to Internet, you won't get to use Internet on your connected PC, if you're having the tethering settings turned off. So that has nothing to do with the USB Storage!
Tip: When android gets the favourable conditions it would itself show the Connect dialog, otherwise you might need to change the port of USB you are connection too. Ory try the might Restart-the-phone trick. This way, you will get the images there! There is no other issue, I have been facing this issue for a long time. Then I caught the errors, the basic one is the USB port, then the Android apps; for that you can restart the phone.
USB drivers are installed by Default while you're connecting the device by general. In Windows OS (That's the only one I use) you would see the popup saying Installing Device Drivers, if that goes perfectly, then you can connect the device. Otherwise the only is this! Or there is a bug in the ROM (if you installed a custom ROM).