Technically, the 'router' you get from the store.. ie your Linksys or what have you, isn't even really a router.
A true Router will route traffic to locations. The one at home only sends it out to your ISP's router which then routes it for you.
So when you ask the difference? Nothing. Marketing. I believe all the routers out there on the market for home use are NAT 'routers.' This also acts as a Firewall as it hides your true internal IP address.