Disconnected after connecting to VPN when using remote desktop

What's happening is that you're effectively cutting off the IP route from the server to yourself - hence the RDP session loss. You can fix it by setting up the VPN in a way that it's bound to a second interface (physical or virtual) so that both the VPN and RDP link can coexist. How you do this depends enormously on a range of very detailed configurations that we don't know right now, so if you want help with this you'll have to come back to us with a LOT more information, just as much as you can please.

Given that you can connect by IP address, the problem is one of name resolution and is likely to be a NETBIOS/DNS issue

At work, your machines are probably on a domain such as "company.local" so when you connect to, say a machine called WORKSTATION its FQDN (Fully Qualified Domain Name) is actually WORKSTATION.company.local

At work you will likely have a fully operational DNS/WINS/NETBIOS system up and running so both the short and long versions work

At home, I suspect that the VPN is not picking up the correct name servers.

If you try connecting to e.g. WORKSTATION.company.local i.e the long name, does it work? If so then its a WINS/NETBIOS problem but if not it is a DNS issue