Green usually indicates you have a device connected
Amber usually is used for traffic between devices
Depending on what type of Modem you have it may only support 1 direct device vs your Router that supports multiple devices. My modem won't issue an IP to get to the internet when I have more than 1 device connected but, after it's booted I can connect a secondary device to manage / view logs on the Modem.
What is the exact make and model number of your "modem" ? FWIW, there is a difference between a stand-alone cable modem, a stand-alone router, and a combination modem/router gateway device. Many people these days are calling the combo gateway devices "modems". To avoid confusion, these terms should not be used interchangeably as they are currently are.