I am using AnyConnect 4.4.03034 on a Mac (10.13.1). I've had zero problems in the US (my home network) and wifi at two hotels in Germany.
Here in Shanghai, a successful VPN connection is hit-or-miss. When I connect to the hotel's wifi, I have access to the internet (anything but Google, as their services are blocked by China), but AnyConnect thinks I am not connected. I retry launching the VPN several times and eventually, I am successful. It does not seem to matter which VPN server I select...
So my first question is does AnyConnect use any sort of Google services? That alone would explain the problem.
Also, as an FYI, my connection to this forum (supportforums.cisco.com) is also very slow, but eventually works (I am entering this New Discussion on the hotel wifi).