The difference is in the pickups.

Most electric guitars use passive pickups that use the interraction between the steel strings and the pickup magnets to induce currents in the coils. This signal is what the amplifier works with.

An acoustic electric guitar usually has a piezo pickup built into the bridge. These pickups output is quite low. So these guitars often have a preamp built into them. Since this type of pickup directly senses the vibration of the bridge saddle, there's no need to use steel strings. They work just as well with bronze or nylon strings. Which is why this type of pickup is the most common type found in acoustic electric guitars.