What's the difference between the 'park' in an automatic transmission vs using the hand brake?

The brake puts a brake on the wheels, like you intuitively expect.

"Park" actually just drives a metal post through your gearbox so the gears can't move. Since the default setup on automatic transmissions at rest is for the gearbox to be connected to the wheels, this should stop anything moving.

This is why, when you put on the parking brake, the car doesn't move when you let off the normal brake. But when you let off the normal brake after putting it in park, it lurches a few inches down the hill as all the slack in the drive train gets caught up [the same mechanism as drive lash].

You should use both. If nothing else, than because the odds of both failing is infinitesimally small, while the odds of one failing is slightly less infinitesimally small.

As has been pointed out a couple times, there's other good reasons for using both:

• Putting the handbrake on before the lash is absorbed puts less stress on the drive train and transmission.
• It's designed to hold your car while it's parked, that's what it's purpose is.
• Apparently some cars use it to automatically adjust the main brakes