The stratosphere cycles through much slower than the troposphere (there are very few clouds, for example). That means that if you put water there, it can stay there for a lot longer, increasing the total amount of water in the atmosphere and warming the climate.
I'll also say that we expect water vapour to increase as temperature increases, which is a positive feedback in the climate system,increasing the warming beyond that of CO2.this is known as the water vapour feedback.