well.... for one, there's the semi-obvious reasons. say you have like... 3-4 or more computers. you can have them all running off the same broadband connection without having each computer rely on each other to stay connected to the internet. the connection is also being split from one connection into multiple connections, which as far as i can think of provides a better transfer rate than having each computer sharing the network/internet connection. it's actually kind of hard for me to think of benefits of a router.... cept for wireless routers.... which there it's pretty obvious. actualy made this post on my lappy over the wireless connection. at first i didn't think i needed one...... now i don't think i'd feel comfortable without one.
1) Sharing a single connection among several machines - other options include paying your ISP for multiple IP addresses (expensive) or leaving one machine on all the time and using it as an ICS host to share the connection with other machines (inconvenient).
2) Security - a router acts as an intermediary between the internet and your local intranet. Unless you explicitly set up port forwarding or DMZ, incoming packets are dropped/filtered by the router. With such a setup, you don't really need a software firewall unless you require outbound traffic monitoring (to check for trojans and malware that attempt to make outbound connections).
If you have more than one computer to be hooked up and share an internet connection you need a router, if you have two or more computers and you want to share printers or external harddrives you need a router. If you ar connected to the internet and you want a hardware firewall you need a router. And if you want to go wireless around the house then you need a wireless router.
Routers are only to route traffic from one network to another. DSL/Cable routers are actually multifunction devices running embedded linux/bsd most of the time which is why they will crash and need a power cycle every now and then.