The easiest way to understand it is that the modem brings the internet to our home and the router takes it and divides it among the different devices that are connected to the network.
An important reason why we should know these differences is that very likely from time to time we will have to test one of them when they present failures, so we must know which is which.
Although both work together to perform the same task, both fulfill different roles, and the truth is that to be secure enough, internet networks will almost always need both.