In rockets, it's stored cryogenicly and the tanks are continually topped up until just before launch. Then the flight lasts a few minutes. Not really applicable to aircraft flying for many hours. Watch a liquid hydrogen rocket take off and see the ice falling off the sides. That ice would prevent an aircraft from taking off if it was on the wings.
As with all alternatives, hydrogen runs in to the simple issue that "oil" is just about the perfect energy source. It's energy dense, easy to store requiring no special systems, easy to handle, easy to control in combustion.