As far as I can see, NAS and routers with a port are aimed at different audiences.
Good quality NAS devices usually have real SATA or PATA ports. Many support hot-plugging and many support various kinds of RAIDs. They also support higher number of disks. 4 disk NAS devices are very common and there are lots of devices with even more ports. On the minus side, they are usually going to be more expensive and will be one more device on the list of devices which will constantly use power and ports.
Also good quality NAS devices are usually fast. They may also have firmware with "advances" features already included such as FTP server, bittorrent client, some fancy software for synchronising NAS with windows and so on.
On the other hand routers are usually aimed at users with one or two external hard drives and almost exclusively use USB 2.0 for connection. They often do not support RAID (and there's little point, except for redundancy, because USB 2.0 can't sustain high transfer speeds) and probably won't have as good out of the box software support as a real NAS device. Their plus side is that you'll save some money, because you can buy a really good router with USB ports for the price of a good NAS device. Some routers may be used with third party firmwares such as DD-WRT or OpenWRT which will allow use of some advanced features such as FTP servers and similar, but will require more time to configure.
In between those two sides you have cheap NAS devices such as external HDDs with network ports, one or two HDD NAS devices which use USB connection and similar.
The problem with them is that their main positive side is that they are cheap and small. They often have worse performance than larger NAS devices and may have performance which approaches performance of routers with USB ports. They probably won't have as good software as more expensive NAS devices, but will have all of their bad sides, except for price. If you go that way, you could probably buy one mediocre NAS and one mediocre router for price of one good NAS device or one good router.
As for which is better, that's for you to decide. I myself went to the direction of router with USB port and OpenWRT. It took some time to learn how to configure it and I don't have great performance of a real NAS, but it was much cheaper option. If you decide to do the same, take a look at DD-WRT compatibility database. What you should be looking for are devices with lots of RAM and flash and as much USB ports as possible. Even if you're not going to use third-party firmware, those parts of router specification are important for good performance and are usually left out on manufacturer web sites.