The benefits of adding seaweed extract to your fish tank(s) include the knowledge that your tank has a great organic fertilizer source that (as long as you choose an organic extract) is free of additives and other harmful components that most prefer to avoid. However, the primary purpose of adding seaweed extract is to balance minerals and vitamins not found within your fish food, therefore check to see if your fish food does not already contain sea elements negating your need for additional extracts. At any point while adding extra fish or new plants you can find that mineral deficiencies will occur and seaweed extract is a good way of fixing those without having to resort to chemical substitutes instead. Some seaweed extracts also purport to prolong the shelf life of crops after harvest through their use but such is generally unsubstantiated though many studies have indeed shown an increase in crop yield from adding seaweed to their tank system. Studies have however continuously shown that the addition of seaweed to any tank will increase the plant anti-oxidant activity protecting the plants from stressors and free radicals.
The disadvantage of using a seaweed extract is that you are adding yet another component to your fish tank(s) and disrupting its natural balance. In addition you are also not addressing the problem, that possibly your fish food is not complete enough for your needs and that you may need to consider a better quality or different food source. Another disadvantage of using a supplement is that if you are not keeping a close eye on your tank you may over supplement your tank, and risk giving the fish poisoning for having an over-abundance of iron from the addition of the supplement. Seaweed itself, being a natural organic crop is not bad for your set-up if you plan on adding it to a salt-water or brackish tank in your system, this means that an extract is wholly unnecessary since the seaweed itself can become another crop instead of just adding an extract to temporarily fix the deficiency problem.
To conclude seaweed extract can add beneficial nutrients to your aquaponics system and generally is a great addition to your feeding line up, if your system needs it. Be sure to check your fish species and food needs before deciding to add this additional iron supplement as you may not even need it and could do more harm than good if used improperly.