Aquaponics is a specialization within the field of hydroponics. The short, and simple, way of explaining the differences between the two is: aquaponics uses fish that provide the nutrients to the plants, and hydroponics uses a chemist to add the nutrients for the plants.
Plants need 16 different nutrients to grow. In hydroponics, these 16 nutrients, along with over 15 other ingredients, need to be mixed by a chemist to create a solution. In aquaponics, all 16 essential macro and micro nutrients and everything else the plants need occur naturally (with the exception of adding iron) by replicating processes from nature. The result is superior nutrient quality and better taste from a well-designed aquaponic system.