They are the pretty much the same thing really, except the Router can have more than one PC connected to the Internet, at the same time!
The Modems are usually connected to a PC via a USB socket, whereas the router would normally be connected using an RJ45 network cable into the PC's network adapter/connection and/or via its wireless technology, i.e. if the Router & PC(s) are capable of both. Some Router models are cable only, with no wireless capability so would need the RJ45 connection!
If capable of both cable & wireless, many Routers can be used in both wireless & cable modes simultaneously or separately thus allowing multiple PCs &/or laptops and any other wireless enabled technology to connect to the Internet or the Network.
Routers were/are designed very much for Networks whereas the Modems were/are usually designed for non-Networked systems such as in the home where one or two PCs would be the norm, though some can have more connections depending on the model.
Single Modems nowadays are becoming much less used and Router/Modems are pretty much the norm because of their greater versatility and their wireless capability, which is very much the current trend.
But they both do pretty much the same thing really in that they enable the connection of PCs et.al. to the Internet one using Broadband technology via an RJ45 cable and a telephone line through an ADSL splitter/filter and the other using a little less set-up!
But they are both essential 'Modems' which stands for: Modulator/Demodulator hence the term Mo-Dem! i.e. they convert tiny electrical signals into something that a micro-processor can handle.
As I said at the start there's not much difference, but I had to elaborate a little to clarify for some! Strictly speaking they should be called Router/Modems not just Routers because that is what they do! But because the human race like to be brief, they've ended up being known as just Routers!