Well, here's a few things to consider:
1) The NAS will run at the speed of your network. This is probably about 100mbps
2) It does not require a computer to function and usually includes utilities such as FTP access, windows file sharing, web management, etc.
3) A USB drive will operate much faster than the network.
4) A USB drive will require it's host computer to be on in order for it to be accessed
5) The host computer must share the USB drive
6) No (well, not really) web admin for the USB drive. FTP access would be provided by the host computer.
So with the above, you need to determine if speed is a concern. I have both USB drives and a NAS connection. It takes about 2 hours to perform a backup to the NAS device whereas the USB drive gets it done in about 20 minutes. (Both have the same drive in them.)
With the NAS, yes you can open it up so that it's available through your firewall and accessable from the internet without having to have a computer turned on. You can also do this with the USB drive, but again, there must be a computer at home to host that USB drive.
If you must go with the accessability from outside your network, I'd go with the NAS. Otherwise, I'd yield to speed and go with the USB.
***Added info ***
In order to access it from outside your network, you'd need to modify your router and firewall settings to create what's known as a DMZ. Basically, you'll allow the IP address of the device to be reachable from the internet. How to do this depends on the equipment you have as they're all different.