Well, first keep in mind that if you cooked meat at 150 degrees it would take a really long time. But it would cook eventually and the same thing would certainly happen to you if you didn't have any sweat glands. I remember listening to a pretty crazy Radio Labs episode about some guys in the 1700s who had this exact same question and they tested it by sticking a wood-burning stove in the middle of a small room and feeding the fire until it was really hot (I think they got up above 200 degrees) and then they just walked on in there to see what would happen. And nothing did. It's pretty remarkable how much sweat can cool us off. If you think of heat as energy (which you should because it is) then it's easy to see how sweating keeps us cool. There's water on our skin, and then one by one the individual water molecules take a bit of our energy (in the form of heat) and uses it to fly off away from us. Since it took energy away from us, we end up a bit cooler. It works great until you run out of sweat, and then I guess that leads to your second question.
So I guess at first it would just smell a lot like sweat, but after a while I imagine it would start to smell a bit like a barbecue pit, which is to say heavenly. I guess the take-away from all this is that if you're going to carry out this experiment to its full conclusion, please think of your neighbors and sprinkle on a little dry rub first.
I found a link to the Radio Labs segment by the way: http://www.npr.org/templates/
