While both Modem and Router play their own (complimentary) roles in providing internet connectivity to your home/office network and devices, there is a difference between these two types of devices.
Hence, let us go ahead and take a look at Modem, Router and the main differences between these two types of devices.
Role of Modem & Router in Internet Connectivity
In a typical home/office network, Modem can be described as the starting point of internet connectivity and Router as the distribution point, from which computers, phones, printers and other devices obtain network connectivity.

As you can see in above image, the Internet Service Provider (ISP) has brought internet connection (via cable, fiber or telephone line) to Modem; which in turn is relaying the connection to Router for distribution to other devices.
Currently, most Internet Services Providers (IPS) provide a gateway or single box; capable of functioning both as a Modem & Router.
The Router gets internet connection from the Modem and creates a Local Ethernet/Wi-Fi Network, which can be accessed by computers, phones, printers and other devices.
Defining a Modem
The term Modem stands for modulator-demodulator, which can be generally described as a box like hardware device, capable of converting incoming analog signals (from the service provider) into digital data; which can be understood by connected devices (computers, phones & others).
Modem connects to the network of your Internet Service Provider by using coaxial cable, telephone line, Fiber-optic cable or satellite, depending on whether you have a Cable, DSL, Fiber or Satellite powered Internet connection.
Defining a Router
Router can be defined as a box like networking device, capable of receiving digital data from the Modem and creating a Local Area Network (LAN), which can be used by your computers, phones & other devices to communicate with each other and the world-wide-web.
When Routers first appeared in the market, they only had multiple Ethernet ports that allowed devices to connect to a Local Area Network (LAN). The current generation of Routers can provide both wired and wireless connectivity to multiple devices like Laptops, Tablets, Printers, Phones and other smart devices.
Gateway Devices (Combined Modem/Router Units)
As mentioned above, most Internet Service Providers (ISP) offer combo devices that function both as Modem and Router. However, you can still use your own Router by plugging the Router into one of the Ethernet ports of the combo device.
This offers you with more control (parental Control Settings) and better performance, especially if your own Router is more advanced compared to the ISP provided Modem/Router combo.
Key Differences Between Modem and Router
As you must have understood by now, the main difference between a Modem & Router is that a Modem is responsible for bringing the internet from the Service Provider to your home/office and Router is responsible for distributing the internet connection to your computer, phone, TV and other devices.
The Modem by itself can bring and provide wired internet connection to a computer. The Router cannot bring internet connection from the ISP, it can only distribute internet connection from the Modem to other devices.
Primary Function: Modem connects your home/office to Internet Service Provider and Router connects your devices to each other and the internet.
Network Type: Modem makes use of Wide Area Network (WAN) and the Router relies on Local Area Network (LAN/WLAN).
Wireless Capability: In general, Modems are not equipped with wireless capability, while Routers are capable of creating wireless Wi-Fi network.
IP Address: Modem receives one IP Address (Dynamic or Static) from the ISP and Router assigns private IP addresses to other devices.
Data Screening: Modem receives data from Internet Service Provider and transmits that data without screening it. The Router in most cases is equipped with firewalls to screen the data, before transmitting it to other devices.
Number of Ports: A typical Modem comes with 2 ports, one of which connects to Internet Service Provider and the other one connects to the Router.
Routers come with 2, 4 or 8 ports for enabling wired connection to multiple devices in addition to providing wireless connection via the Wi-Fi Network.