20190608 A modem is a modulator/demodulator and performs the task of encoding/decoding a digital signal from/to a waveform signal that is typically sent over telephone wires. It is the modem that establishes a connection to the Internet for us. On the other-hand, a router decides how to route digital messages between computers in a network based on the addresses contained in the messages.
On installing a new modem/router be sure to set up its security. The key is to ensure that WAN login is disabled (so that no one can login remotely to the router, though this is likely the default) and that the default password for the router is changed. You might also consider changing the SSID (service set identifier) and the default router IP address as additional measures to remain secure.
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