Except for the gas vs electric, what’s the difference in an E-Bike and the old MoPed...
Bell Bottoms
Ebikes are (generally) much less power than a moped. The Lemond bike mentioned above has 250 Watts output where an old Puch Moped was 3hp, roughly 2300 Watts.
I am researching an ebike buy for myself. It appears places that allow ebikes on multi-use trails/sidewalks limit the bikes to class 2. That is 750W max with a power dropout @ 20mph (no power assist >20mph), and pedal assist only (no separate throttle).
Class 3 ebikes are for off-road only and have 1000W motors, a max speed of 28mph + with motor-only driving (throttle).
There are a few bikes (I think these are Class X) that have AWD dual motors and are pushing the +2000W threshold. Fat tires and a large cargo rack made specifically for hauling you in to your deer-stand (no scent trail) and back out with a deer on the rack.
Most of the controllers can be "jail-broken" and programmed to allow the motors to run at their max - and you can also buy professional looking Class 2 decals to put on a bike that might be something other than a legal class 2.