Unable to open "Cisco Anyconnect Secure mobility client" on my Ubuntu 18.04 LTS after the complete installation

I had the exactly same problem, the VPN was working but couldn't see the GUI, I found that is because of some missing dependency (libpangox-1.0-0 and libcanberra-gtk-module), try installing it:

sudo apt install libpangox-1.0-0 libcanberra-gtk-module

You may check if you need anything more or have other errors by running the UI executable, mine was installed on

/opt/cisco/anyconnect

You can run bin/vpnui and check what error you may have, hope it helps!