When trying to get Anyconnect VPN working after upgrading from Fedora 26 to Fedora 28, vpnui hung up and refused to connect. The solution was to copy the /lib/libstdc++.so.6 file from Fedora 26 an put it in ~/.cisco/hostscan/lib and then it ran fine. I did this for the root user as well.
10 October 2018 – Update
Bugzilla report http://bugzilla.redhat.com/show_bug.cgi?id=1576910 shows a problem with NetworkManager and its interaction with Anyconnect. I can now confirm that uninstalling NetworkManager-config-connectivity-fedora seems to make Anyconnect stable on Fedora 28 with Anyconnect version 4.603049
Might be related to https://bugzilla.redhat.com/show_bug.cgi?id=1576910
Thank you for this update Michael. It is a different problem. In my case, Anyconnect failed to get past the login without the fix that I posted. This seems more related to stability of the link at DHCP renewal time. Strangely, this has not been a problem for me, but one of my colleagues has seen it, so thanks for the information.
Michael