The (probable) reason you cannot access the server through your laptop is because of DNS not resolving. It is trying to connect to server via an external IP ( the one connected to internet) and of course, that won`t work.
If your server is connected to Internet, it is most probably assigned and accessed via a Domain name, ie: server.mycompany.com.
This server should (usually) be accessible by 3 methods: the DNS name, an external ip, or an internal ip. This last method is what we need to access locally.
If it is a web application you need to access, simply using the internal ip of the server would suffice to access it:
Instead of server.mycompany.com, one would use the ip, ie: http://192.168.0.23
To avoid your laptop being disconnected from the server when internet is down, you could add an entry in your host file that points to the local ip, IE:
Of course, this means when outside your local network, you would need to remove this entry to allow the DNS to resolve through the external ip / interface.
Hope this helps, good-luck!