Well you have to remember that the line many times is pretty gray between the two. While both can perform the task of routing and firewalling it comes down to the products intended focus.
In your example for a standard "home user", a standard gigabit router vs a Cisco ASA (Adaptive Security Appliance) appliance (firewall), the home user would probably never know or understand the difference. They are both able to route and firewall packets very effectively at a high speed.
Where the specializations start coming in are in their intended purpose:
A dedicated router has the ability to learn and propagate routes to neighboring routers using protocols such as BGP and RIP, and perform these functions at very high throughput.
A firewall or security appliance focuses on security and may not support advanced routing protocols especially in larger environments. They will focus on connection and state tracking over throughput or routing. Many firewalls offer two modes: "routed" and "transparent", transparent being like a switch at layer 2. A security appliance may also focus on VPN connections including remote access clients; it may also add functionality to include additional features like content filtering, Intrusion Prevention, antivirus, etc.
Now to answer the question of "why spend thousands on a Cisco ASA vs a Linksys with gigabit ports?"... the answers there are simple: stability, capacity, support, and feature sets.
For example, an enterprise Cisco ASA has hardware crypto accelerators capable of terminating 3000+ VPN tunnels concurrently, providing SSL based VPN services to end users authenticated against active directory, handing URL's off to Websense for URL filtering, high availability in the event of hardware failure, firewall access lists with hundred's of lines, the ability to parse vlan tags for interfaces (might have 50 different networks coming in over 1 physical cable), the ability to create multiple contexts which allow you to create a separate virtual firewalls each of which may have it's own admin users... best part is that it can do much of this all at the same time. (not all of it though, there are limitations with some configs) Those are some of the nice features, but you also have to remember that the devices are built REALLY well and unlike consumer products they rarely fail, and if they do happen to fail, you can call Cisco who will have a replacement to you in record time, or great INTELLIGENT tech support to help you fix whatever you broke with your configuration.
In comparison, a belkin or linksys will probably max out at around 5 - 10 VPN tunnels with maximum throughput usually in the 15 - 20 mbit range, firewall rules are extremely limited in comparison both in numbers and in functionality. You'll also find that it may need power cycles every few weeks to resolve "connection problems". If you need tech support you can try emailing tech support which will likely just keep asking if you can ping the box for a few days before you finally give up and just buy a different brand. The "DMZ" feature also is extremely dangerous and doesn't accomplish anything like a real DMZ will accomplish, in fact it just exposes your whole internal network, just like UPnP can do if one machine happens to get infected with something. Also just because the belkin has gigabit interfaces doesn't mean it's capable of actually <doing> it. Cisco tends to under rate their specs, where as most of the home ones do a nice job of avoiding talking about them. 🙂
Sooo... in short, while a belkin or whatever does a good job of fitting the needs of home user, it really doesn't belong in an enterprise environment where uptime, supportability, government and PCI compliance standards, performance, and overall feature sets aren't just "nice to have" but are absolutely necessary. As the old saying goes, "you never get fired for buying Cisco".
Keep in mind the above was in reference to Cisco (since that's what you asked about), but much of the same applies to many of the enterprise brands / products from companies like Juniper, CheckPoint, etc...
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