I finally succeeded in getting the Cisco AnyConnect VPN client to work on my Windows 8 home PC. I couldn't get the ActiveX or JAVA-based versions to work, so after logging in to the VPN portal, I got an option do download the software and install it manually.
However, once I tried to connect, I got the following error:
To solve this, open the Registry Editor and go to [HKLM\SYSTEM\CurrentControlSet\Services\vpna] and modify the DisplayName key so that it only contains the text
Cisco AnyConnect Secure Mobility Client Virtual Miniport Adapter for Windows x64
Once this is done, you're all set and the VPN client should work just fine.