This is the main difference between soil and hydroponic gardening. In hydroponics, most of the nitrogen must be supplied as NO3(-) because the hydroponic media allows ammonium ions to become toxic exceedingly fast. For example, hydroponic plants can withstand concentrations of nitrogen (as nitrate) up to about 250 ppm while concentrations of nitrogen as ammonium are only withstood up until about 30 ppm. This is the reason why urea cannot be used as a nutrient salt in hydroponic gardening to supply all the nitrogen needed by the plants.
So if plants in soil and hydroponic media assimilate the same nutrients, why can plants in soil be fed nitrogen as ammonium but hydroponic plants cannot ? The answer is quiet simple. Bacteria present within the soil are able to efficiently convert ammonium ions into nitrate ions, effectively reducing the amount of ammonium the plant "sees". In fact, plants in soil also absorb nitrate, the only difference is that there are bacteria that can convert ammonium to nitrate, reason why nitrogen can be supplied as ammonium to plants present in soil.
So next time you are searching for a nitrogen nutrient for your hydroponic plants, remember to search for nitrate salts as more than 90% of your total nitrogen source. The most important salts for providing nitrogen as nitrate in hydroponic gardening are potassium nitrate and calcium nitrate. This is important to remember, as using ammonium salts to provide your plant's nitrogen will ultimately kill them in hydroponic media ! (below, an image showing the effects of ammonium fertilizer in hydroponic plants)