Hi sptham30,
You may want to setup port forwarding in your router to map the listening port of NR server from your private IP to the public, so that NR clients can access to it from anywhere.
If you don't have the access to your router, you probably cannot use NR Free/Pro. However, you can use NR Mesh service, which is a cloud VPN service and the servers are hosted and maintained by NeoRouter team. You just need to install NR Mesh clients and run.
http://www.neorouter.com/mesh/In your case,
1.NR server keep updating with our backend server to sync your current public IP. That's why you can see the IP changed.
2.IPchicken.com gives you the IP of your current client. If your current client is in a different LAN from NR server, you will see different public IP.
Let us know if you have any question.
KevinZ - NeoRouter team