So from my dictionary 蒸気 means steam, and 水蒸気/湯気 both mean water vapor. What I would like to understand is when is 水蒸気 used over 湯気 ?
Scientifically speaking, 湯気 does not mean vapor but steam because 湯気 is a visible thing whereas vapor is invisible gaseous thing. I think 蒸気 corresponds to vapor in English, but we don't usually distinguish between them same as in English so there is not much difference between 湯気 and 蒸気.
Setting aside above, 湯気 is more common phrase in spoken language than 蒸気. When there is hot steam coming out of the kettle, we are more likely to say 「やかんから湯気が出ています」 than 「やかんから蒸気が出ています」. 蒸気 is used in academic phrase or specific terms like below:
・水蒸気爆発 (steam explosion)
・蒸気機関 (steam engine)
水蒸気 and 蒸気 are mostly the same as with in English - 蒸気 is to vapor as 水蒸気 is to water vapor.
As a side note, 「スチーム」 is also used in electric appliances and beauty equipment like スチーム式加湿器(steam humidifier), スチームアイロン(steam iron) and スチームオーブンレンジ(steam oven).