It still sounds to me you've not configured the routers.
To provide an example.
If I have a home network, and work network and I want to connect to the work network from home, I need to configure the routers or use a VPN software. To provide an example:
Home network:
My Computer (192.168.0.100) <- private IP
My router/modem (192.168.0.1) <- also private IP but is known as 'gateway IP'
My ISP (123.123.123.123) <- the public IP, assigned by ISP.
Then in my work network:
MySQL server (10.0.0.100) <- also a private IP
My router (10.0.0.1) <- a gateway, too.
Work ISP (231.231.231.231) assigned by ISP.
The bridge between two network is of course across the network, meaning one network only see the other's public IP.
If I tried this connection setup:
My home computer (192.168.0.100) --> MySQL Server (10.0.0.100)
what actually happens is this:
My home computer asks router (192.168.0.1) "Where is the 10.0.0.1?", and the router would reply "I don't have any machine here with that address." Thus the connection fails.
Now if I tried to use public IP:
My home computer (192.168.0.100) --> MySQL Server (231.231.231.231)
What happens is this:
My home computer asks the router "Where's the IP 231.231.231.231?", the router will then ask the ISP's DSN server, "Where is the IP 231.231.231.231?", and the ISP DSN will then ask other DSN server until it reaches the work ISP's DSN server which will reply, "here it is." and thus forward the connection request to IP 231.231.231.231.
But the work router (which is listening at 231.231.231.231 public iP) will say, "I do not allow any outside connection, so I'm rejecting and closing this connection."
Thus, you need to access the router directly and tell it to set up port forwarding. The port forwarding would be something like "TCP 3306 Any source IPs ==> 10.0.0.100".
So doing the connection above again, then the work router will be able to say, "Yeah, I have a rule telling me to forward this data to the 10.0.0.100" and it does, finally reaching the MySQL server and interacting with the MySQL.
To configure the router... keep in mind that it is dependent on which router you have, but we'll use a simple router you usually buy from Best Buy as an example. Hopefully you can then figure out how to configure your work router. Anyway, usually, if you buy say a D-Link router, it has a default IP of "192.168.0.1", or maybe "192.168.1.1". If you are uncertain, use the ipconfig and look up "Gateway IP".
Then in the internet browser, you type in "192.168.0.1" for the address bar. It will then connect to the router, and you will be prompted to log in. Usually the default setting is something like "Admin/(blank password", or "Admin/Admin" or "Admin/password". Again, consult your router's manual for the correct default login, if it never has been configured before. Then that's where you set up the port forwarding so it can then redirect all data on port 3306 to MySQL.
Now, you also need to be aware of the risk this places on your work network and for this reason, it is usually good idea to require some kind of secure connection, perhaps by using SSL, or VPN. VPN is different from the above steps because when you open a VPN software & successfully connect to the work router, VPN will then request your work router (10.0.0.1) for a local IP and get one... say, (10.0.0.200). Therefore, your home computer becomes "local" and thus can access MySQL server at 10.0.0.100, without need to go through the public IP.
I hope this helps....