Well, the only main fault you could possibly run into with external HDD's is that the hardware itself may fail. It's rare if not impossible for a HDD to have a software fault because they don't have anything such as no operating systems etc on them to crash (unless you attempt to update the firmware and end up crashing it, however you should have no need to do this). The only thing you'd need to do if it isn't done already is to simply format it.
So the HDD failing due to hardware is the same problem as a burner faulting, although granted the HDD costs more to replace. I find HDD's to last many years more than basic optical drives though.
Virus attacks could be a problem if you don't use common sense with your HDD. Although the same could be said when burning to a CD / DVD. You could end up burning a virus to the CD / DVD and infecting any machine you insert it into. As I said there are very similar pro's and con's for each. All you need really with an external HDD is common sense and you'll be fine. If your worried about viruses, just scan the external drive each week as you would your main internal HDD.
The main problem with an external HDD choice though is cost. They're not exactly the cheapest component on the market. However purcashing an optical burner and the media to burn on each time you run out of blanks all adds up too. Both are good efficient means to backing up data, at the end of the day it's which appeals more to you.