they both do a different job.
A modem connects you to the internet, and a router acts as a distribution centre on a network - enabling networked computers to send and receive data to each other.
I can understand your confusion. Nowadays most wireless routers have a broadband modem built in, so you can use one device to connect and share a broadband service over a wireless network. You can get plain routers, and in that case you would need a separate (Ethernet) modem that is connected directly to the router.
The advantage of using a router is that you can access and share an internet connection without the need for a particular computer to be running. All computers have internet access via the router, and whether it has its own modem, or connects via a separate modem the router will always be connected (as long as it's turned on), and will allow any machine on a network to access the internet.
I hope that helps.