Sultry implies something wet or humid, in addition to the heat, like tropical hot weather.
Stifling implies something that makes it hard to breathe, like hot weather. But this can also be used to describe an oppressive atmosphere in general, not necessarily associated with heat.
So for instance, you would *not* use "sultry" to describe the weather conditions in the desert (like, say, the Sahara) because it is a dry climate, but you may want to use it to describe conditions in the Amazon rainforest during the summer.