What is the difference between an aeroscraft and an airship?
Aeroscraft is their brand name. It does not mean anything generally.

The general terms are:

"Aircraft" (but not "aerocraft") means any machine capable of sustained flight in the atmosphere.

"Aerostat" is a lighter-then-air aircraft. That is it has large shell filled with gas with lower density than air and uses buoyant force on that for lift. This includes balloons, hot-air or gas filled, tethered balloons and airships.

"Airship", or "dirigible" is an aerostat equipped with propulsion that allows it to move on it's own (as opposed to balloons that are tethered or carried by wind).