I noticed that the latest Opera 8 browser also does not work on the 6680. That gave me another angle to explore. The answer is: (drum roll) the 6680 does not want a proxy on the GPRS access point. For some reason my service provider added a proxy server in the settings they provide.
I deleted the proxy from Tools -> Settings -> Connection -> Access Points -> Internet.GPRS -> Advanced Settings.
Everything now works 100%.