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The steam and water can have the same temperature but the steam would carry more energy in the form of heat since it takes energy to drive a phase change from liquid to gas. Assuming that a burn would be stronger based on the energy it transfers onto skin, the steam would have a stronger burn.
If you look at the heating curve of water, there is a much bigger "flat line" going from liquid to vapor than from solid to liquid, which basically tells us that there is more energy required to vaporize. With this higher amount of energy required to vaporize compared to transition from solid to liquid, there is a lot more energy that is being released when steam burns you, which is why the burn is stronger.
The process of steaming converting back into water as it touches your skin releases a lot of heat as a result of the phase change. That transition emits heat which can be seen as heat on top of the pre-existing 100 degrees.
It makes more energy to change from liquid water to water vapor. And as Matthew said, there is a longer flat line from liquid to vapor than from solid to liquid, which basically means that more energy is required to get to that vapor temperature.
The more kJ/mol that the phase change takes, the more energy is released. A liquid burn does not have a phase change; it is merely the water that gets heated, but has not reached the point where enough energy changes the liquid into a gas. By contrast, a steam burn, upon contact with your skin, turns back into liquid, thereby releasing all the energy it initially had to change into a gas and a heated liquid. All that energy that gets released burns skin much more badly than a liquid burn.
Water molecules in steam have more energy at 100 degrees Celsius than the molecules in liquid water at the same temperature for the reason you stated. Even though the temperatures are the same, the steam contains more energy capable of doing work to mess up your skin upon contact.
Visible on the heating curve that we looked at in class, there is a significant amount of energy stored in vaporized water. When this vapor comes into contact with human skin (which is significantly cooler than the vapor) the vapor releases vast amounts of energy to return to its liquid state, which is in fact still extremely hot. Then the liquid water must also cool which gives off another significant amount of energy. The amount of energy released is far greater from its vapor state to its cooled liquid state than simply from its vapor state to its heated liquid state.