What is a modem?
A modem is a hardware device built solely to do one task - to send traffic from point 'A' to point 'B', without working with data. A modem is not programmable, so data security services and traffic routing capabilities are not present in such devices. In addition, the modem functions as a key for Internet access. Like a key, most modems cannot make a connection without the help of software running on the device to which it is connected.
(e.g. TRM link)
What is a router and what is the difference with a modem?
A router, on the other hand, is a more advanced device in industrial networking that allows access to the Internet autonomously, without the integration of software on the connected device. Like a modem, it is used to provide and manage the connection to the Internet. However, when using a router, there is no need to use another device to control it. A router provides the connection to the Internet on its own. In addition, a router also provides the Internet connection to multiple devices, through multiple interfaces. A router has data routing capabilities that allow it to provide internet access to multiple devices at the same time. Managing multiple interfaces and various connected devices makes routers very versatile and adaptable to various industrial environments. Another huge difference between modems and routers is the functionality of the software. Routers are usually equipped with advanced data security services such as firewalls and multiple data tunnelling services. These features provide extensive data security and lower the risk of cyber attacks. All in all, routers are best defined as network devices that provide a secure Internet connection to multiple devices simultaneously.
(link: Firewall and Routers)
What is a gateway and what are the differences to modems and routers?
A gateway is very similar to a router in terms of functionality, since both routers and gateways work independently. Gateways and routers are usually equipped with extensive security features such as firewalls and data tunnelling capabilities. The main difference between routers and gateways is the number of interfaces they can support. A router is designed to handle multiple interfaces, whereas a gateway is usually designed to have only one interface. Thus, it can be said that routers are intended to connect multiple devices to the Internet and gateways are intended to connect only one device.
(link: Gateway and Couplers)
In conclusion: every Router is a Gateway, every Gateway is a Modem, but not the other way around.