Ignoring edge-cases due to messy regulation and manufacturers playing god, the proper normal moped and e-bike differ thusly:
Mopeds rely solely on the motor/engine, and have no pedals. They're heavier, less flexible, more expensive, and harder to maintain/repair yourself. They also go faster, though speed limits are very much down to regulations.
Generally in europe the speed limits are 45km/h for mopeds and 25km/h for e-bikes, but that limit only applies to the motor assist of an e-bike, you can pedal as fast as you're physically able.
E-bikes are amazing because you get almost all the benefits of bicycles plus optional assistance from the motor, which obviously makes it very approachable to people who aren't so keen on exercise.
And bicycles are great because they don't require any sort of license or registration, are cheap to buy, incredibly cheap and easy to maintain and repair (it's quite possible to do extensive field repairs with a couple tools and materials that easily fit in bags), very portable and easy to store (can bring them on most trains), and riding a bike is just generally very pleasant.
Ebikes are basically the perfect personal vehicle for any vaguely urban area. Basically 0 ongoing costs, trivial to store and park, pleasant and good for your health to use, and no negative effects on your environment.