"Modem" vs. "router": tech twins or distant relatives?

What is similar?
Both modems and routers are networking devices used in home and office networks related to internet connectivity.
What is different?
A modem connects a network to the Internet through an Internet Service Provider (ISP) by translating digital and analog signals. A router connects multiple devices within a network to each other and, with the help of a modem, to the wider Internet. A modem establishes the primary connection, while a router distributes that connection to various devices.
Which one is more common?

Examples of usage
Modem- The modem is plugged into the telephone line to provide internet access.
- After resetting the modem, the Internet connection was restored.
- The technician installed a new modem to improve connection speeds.
- The router enables all the devices to access the Wi-Fi network in the office.
- You need to configure the router to secure the wireless network.
- The router distributes the internet signal from the modem to all connected devices.