I know very little--which is to say nothing at all basically--about hydroponics, but my understanding is that it is the culture of terrestrial plants using water without soil. This may seem close to aquarium plant culture, but in fact aquatic plants require somewhat different nutrients, or require some of them in quite different proportions. For example, terrestrial plants require sodium, silicon, iodine, and cobalt for functions not necessary in aquatic plants, such as one obvious example, the fixation of nitrogen from the air. This is one reason fertilizers for terrestrial plants should never be used in the aquarium.
Aquatic plants require 17 nutrients; the macro-nutrients are oxygen, calcium, hydrogen, carbon, magnesium, nitrogen, sulphur, potassium and phosphorus, while the micro-nutrients are iron, copper, zinc, molybdenum, boron, manganese, chlorine and nickel. They require these in very specific proportions to each other; an excess of some nutrients can cause adverse reactions such as the plant shutting down assimilation of specific other nutrients. For example, an excess of potassium will result in a deficiency in iron.
It will also be obvious that some of the micro-nutrients are heavy metals (iron, copper, zinc, manganese) and these are highly toxic to all life--plant, fish and bacteria. While plants have the ability to take up (as opposed to assimilate as in nutrition) these substances, they only have a certain capacity for this.
The form of some nutrients is also critical. Iron in the presence of oxygen changes its ion form such that plants cannot assimilate it; chelated iron overcomes this problem.
All of the above demonstrates that the specific quantity of each nutrient must be carefully controlled. Not only will the plants suffer, but the fish also can be poisoned by excessive minerals. On another level, some nutrients in excess will promote algae.
Some form of fertilization is almost always required, since all the necessary nutrients are unlikely to occur solely from tap water, fish food, and organics. There are two basic approaches to providing these. One is the EI or Estimated Index method that is successful (depending how you view all this) in high-tech planted aquaria that have mega light and added CO2. Dry nutrients are mixed and added to the aquarium, followed by a major water change to remove the excess that the plants can't use and which otherwise may be toxic.
The second method is well suited to low-tech setups, involving use of a comprehensive liquid fertilizer and sometimes substrate nutrients whether as fertilizer tabs/sticks, an enriched substrate or a soil substrates. This is my preferred method, as it basically avoids any possibility of toxicity and with a good fertilizer will provide all that is required safely. For the record, I use Seachem's Flourish Comprehensive Supplement for the Planted Aquarium which to the best of my knowledge is the only preparation containing all nutrients (save oxygen and hydrogen which obviously occur in the aquarium naturally, and nickel which interestingly is not included in any aquarium plant fertilizer I have come across). The nutrients are also in the approximate correct proportions, as shown in the list on their website: