Memory sticks are ok for short term backup or for moving files from one computer to another, but not for long term backup (they can fail).
An external hard drive uses the same sort of hard drives as are used in PCs and laptops, so have much larger capacity.
It is VITAL to backup important files like digital photos, music, documents etc as a computer can crash, get stolen, flooded (if a house is flooded), get a major virus and so on, which could all stop you getting the data off it.
However exteral hard drives can also fail so having just ONE external hard drive is not really enough and it is a good idea to backup your data on a number of different devices in case one of them fails or gets stolen or lost.
Some companies now offer the ability to backup your data on web sites (secure of course) and some companies like Google, Microsoft etc offer so much data for free. This is called storing your data "in the cloud".
All in all, just make sure your important data is backed up.