What are the medical differences between a warm mist humidifier and a cold mist humidifier?

They are different. A warm mist humidifier is a device that heats water into steam and a cold mist vaporizer uses a device to spin and genrate tiny droplets of cool mist like a waterfall does. Both create mist but by different methods. There is debate as to which is a better device. Each has its pluses and minuses.
Safety. Most warm mist producers involve a heating element that can burn down a house or produce scalding when you get up to add water. They are cheaper but dangerous. All mist will cool before it gets to baby anyway. They have a place in controlled hospital settings when used to moisturize oxygen at the source, but none in the home.