Steam is wet, Sauna is dry.
Steam is typically installed in your shower or other waterproof environment and all you see is the control and the steam head/outlet. A Sauna is a room constructed of wood, generally cedar, where a heater with stones is visible in the room. Accurate Industries does a nice job of summarizing the basic differences in this article www.accurateindustries.com/blog/?p=125
Now, you can bring a little steam into the sauna environment by ladling water on the heater stones but generally a sauna room is a very dry environment. As for your asthma, this is a chronic disease which requires strict and personalized symptom management so this question should be directed to your doctor.