The only reason I would use an external disk is if I planned on taking the disk with me some where, or otherwise needed the portability. If your data disk is going to stay with that computer, then I would go internal, period.
If, on the other hand, you do need to move the entire data disk around, external is your best option.
In the middle is if you need to move partial data from this disk, in which case (unless it's a very big part) I would go with the internal disk, and then get a smaller external disk or else rely on network/internet to share the data.
On the reliability front, external disks typically have an extra risk factor of being dropped, something internal disks (except those in laptops) generally do not face; barring that, both are, generally speaking, equally reliable. Additionally, unless you are using eSATA as your connection, it is difficult if not impossible to monitor the S.M.A.R.T. status of an external hard drive, so it could more easily fail on you without warning (of course, if you're not monitoring S.M.A.R.T. on your internal disks, this doesn't change anything, but you should be).
Backing up a disk is, for the most part, wholly independent of its external/internal nature. The only caveat here is that there may be a manual component required for an external disk if its drive letter/device letter changes when it is plugged back in after being disconnected.