How can I make Windows remember my Galaxy S3
You either have to forget the network connection,by navigating to Settings >> Network and Internet >>  WiFi >> Manage known networks, and choose "forget" for the appropriate network, then reconnect and see if problem persists or unistall the device driver.

If its USB tethering try unistalling the Network adapter:

Go to device manager >> Network Adaptor>> Right click on your wireless adaptor and select uninstall. (Also enable "show hidden devices" and delete old instances of this network adapter, its usually RNDIS Device)

Reboot and reconnect your device. Sometimes its just occurs because of some corruption in registry.