The hydrogen bomb has absolutely nothing to do with the chemistry of hydrogen. Hydrogen gas does burn in oxygen, forming H2O, and the process releases a lot of heat. However, the reaction occuring in a hydrogen bomb is a nuclear reaction, not a chemical reaction, which releases a staggering amount of energy, unimaginably more than just burning some hydrogen gas.
A hydrogen bomb contains two isotopes of hydrogen, called deuterium and tritium. These both have only 1 proton, but deuterium has one neutron and tritium has 2 neutrons. Normal hydrogen has no neutrons, and 1 proton. The chemistry of tritium and deuterium is almost identical to that of hydrogen - tritium will burn in oxygen to form water with two tritium atoms instead of normal hydrogen.
What happens in a normal atom bomb is that a very large nucleus is split apart, and this releases energy. In a hydrogen bomb, two small nuclei are joined together, which also produces energy. The reaction that occurs is:
Tritium (1 proton, 2 neutrons)+ Deuterium (1 proton, 1 neutron)-> Helium (2 protons, 2 neutrons) + 1 neutron + a lot of energy
The nuclei are actually joining together to make another element. When you burn hydrogen, the nuclei are not changed, but the atoms are joined together in a different way.
Because both nuceli are positively charged, a lot of energy is needed to get them close enough to join to make helium, which is why hydrogen bombs contain uranium or plutonium, to create a relatively small conventional nuclear explosion which releases enough energy to get the hydrogen to fuse, causing a much larger thermonuclear explosion. This can't possibly happen at room temperature, or even a hell of a lot hotter, so you can have a balloon full of hydrogen without any risk that it will create a nuclear explosion. It could explode if you set it on fire, but that's a chemical reaction, not a nuclear one, and it produces probably billions (maybe trillions, it's a wild guess) of times less energy.