The greatest benefit to an external drive is that it is portable. When you back up your files to it, you can store it at another site.
Think about it - if your home or office were broken into and your computer stolen, a second, internal hard drive would do you no good - it would be gone, too! The same with damage/destruction from fire, flood, vandals, hurricane or other disaster. The only thing a second, internal drive protects you from is failure of your master drive.
With an external drive, if some disaster happens, you can be up and running with your important data as soon as you can get to a local store for a new computer. (Note: you should also burn duplicate copies of the install disks for your software and store THE ORIGINALS somewhere else, too.) Off-site storage of your data is a MUST if you do anything other than play games and send family e-mail.
Backing up to CDs or DVDs is an option, too, but only if you DO IT. Knowing my personal habits, I know I wouldn't do it on a regular basis. Since our small business (commercial printing) may generate or alter a hundred files a week, I'd have far too much data at risk.
Here's how it works for us. As we create or alter files, we save them to our internal drive in a folder named THIS WEEK (with sub-folders for each customer or catagory). One day a week, I bring the external drive to work and we add the files in THIS WEEK to the backed up archive of all our files. I make sure to take the drive home with me at the end of that day. To be really efficient, each worker could back up their THIS WEEK folder each day onto a jump drive and carry it home with them, but we haven't gotten that far yet.
If you think the CD/DVD method would work for you, fine. If you already have a burner and don't have several gigs of data to back up, this can be the easiest and least expensive way to go. Just remember that burns aren't always successful and disks can break, scratch, get lost, etc. As for keeping track of what files are on which disk, that would be a nightmare for me (business), but maybe not for your usage. Yes, we've had a removable drive fail, but all the orignals were still on our computer, so we could create a new backup to a new exernal - nothing lost.
There are pros and cons to each method. The BEST method is the one that you will do regularly and store off-site. You need to determine what works for you, determining your level of computer usage, amount of data to store, and your personal habits.
Hope this helps, Jane