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Yep, exactly what @Nocturnal Thinker said, the modem is pretty much how you get internet, while the router is essentially how you transmit it for consumption by WiFi-enabled devices. Both are vital, for sure!
I had learned about the modem and router when I was working in the bank in 1999 up to 2003. The modem is actually the Modulator/Demodulator which converts the round waves from the source into the square waves that computer can understand and vice versa. The router is just like the electrical extension sockets or surge protector where you plug it to an electrical socket that will provide more sockets. However, the router has a maximum number of wifi or wired users although for the home I think the maximum is sufficient.