For me the difference is all about dynamism - paragliders float and hanggliders swoop! Before I had my first trial flight on a HG I did a couple of paraglider lessons - it was alright but I didn't find it very exciting. Once you're up you essentially float along in your comfy flying armchair at a fixed speed, with the ability to steer left and right (although when it gets thermic I believe there's a huge amount of feel and finesse to learn, which keeps things interesting). Most of the challenge seemed to be getting off the ground in the first place (launching the wing, which can lift 6 times your bodyweight, without it dragging you through a hedge).
Luckily I didn't finish my training and spend loads of money on kit, because when I later had a go on a hangglider I knew that was the one for me! The key difference is that a hangglider has pitch control, meaning you can dive, swoop, carve through turns and wingovers, loop-the-loop (if that's your thing) and burn down through turbulence without sitting there dumbly sucking it up. And, lying down on your front makes you feel like superman!
Not to say there's anything wrong with paragliders. If you plan hiking anywhere with it they're definitely the wing to choose, and they are probably less work on the arms to fly. And, strangely, they can do much more radical aerobatics than a hangglider (lots of end-over-end tumbling, which we try to avoid if possible
), but I just like the feel of flying a hangglider - it's the closest thing to how you dream of flying, except for maybe wingsuits.
Here's some vids I like of both sports, which seem to capture (for me at least) a bit of the essence of each