I would say it's condensation. And the air humidity is pretty high, after all that's what saunas do...
Basically stones don't radiate heat into the air that well. Air is mobile, but not a good conductor. But, liquid water gobbles up heat pretty fast. It becomes steam, which is mobile and spreads around pretty quickly. The air becomes saturated with steam/hot water droplets. These are at a higher temperature than you body (or atleast nearby). They condense on your skin(saturated air=humidity=condensation), releasing latent heat. It feels hot. There may be some more heat due to temperature difference, but I can't comment on that as I don't know exactly what temperature the vapors are at. Since they can be both vapor and steam(as far as I can tell), I don't know nwhat temperature they ought to be at.
Regarding the sweat, IIRC the sweat doesn't evaporate in a sauna, though we do sweat a lot. In the end, I just think that it drips off. In humid conditions that is the issue with sweat. It tends to accumulate as it can't evaporate. Pretty annoying if you live in a tropical area.