Routers are a basic computer dedicated to the task of shifting those packets from A to B through the shortest/fastest route. An old PC by no means is faster than an enterprise router. However, the little plastic boxes that claim to be routers for the consumers are no more powerful than good phone from a couple of years back.
By all means this is perfectly fine for the "average" user, as most the majority of people are just connecting their computers to the network and going to the World Wide Web.
However, when you start getting alot of traffic, a faster solution such as an old PC, will start to beat the crap out of a cheap plastic box.
Say you have multiple users and one is doing some gaming, another a VOIP call and the third is just downloading porn off the internet. Obviously the most important traffic is the gaming traffic and VOIP as delay in processing those will cause lag (gaming), stuttering in VOIP, especially if the third person is hogging all the bandwidth for his busty babes film.
A cheap router will just process everything hoping it will be okay. Although the VOIP call could be dropping out and the gamer could be getting kicked off of servers for too high a ping. With a dedicated machine (or enterprise routers) you can set limits and priorities. So say no single machine can use more than 80% of the avaliable bandwidth. That will save the gamer and VOIP call. Then you can add QoS (quality of service) which ensures that the gaming and voip traffic has a higher priority over the "two girls one cup" film being downloaded. This along with reducing response time will greatly improve the experience for the VOIP.
Plus those cheap routers are trying to be multiple things. A DHCP server, DNS, Time server (sometimes) whilst still trying to move packets. Theres only so much that can be done with only a 400Mhz core clock. A machine that is faster will handle response to a DHCP much faster without causing any issues with traffic.
They can also greatly improve internal network performance. Back before I built my own router (a good few years ago) if I tried to watch anything off of my NAS. Everyone else on the network wouldn't even be able to go on the internet, despite the fact I wasn't using an external connection.
I could seriously go on all day, but if your only have a small network (like 1-4 computers) and dont have internal servers running. Okay you wont really benefit from a custom router. However if you have people playing games, doing VOIP calls whilst downloading porn and moving files to and from a NAS/ other PC. Then you will seriously benefit from the extra grunt.