There are limited ways your body can cool itself. They depend on natural laws of chemistry and thermodynamics. When the ambient air temperature is above body temperature, you can't radiate heat to the environment anymore. Even at 90 F, the radiation from you to the surroundings is slowed considerably.
When humidity is high, sweating (the body's most effective mechanism of cooling) becomes ineffective. The water has to evaporate to do much good; when it boils, it removes energy (heat) with it. When it just sits on your skin, it isn't helping much. And, just because you don't see sweat on your skin doesn't mean you aren't sweating. It is just doing its job of evaporating.
When heat and humidity are both high, it can be dangerous to do extended bouts of exercise. Your natural mechanisms simply won't work very well. I seem to remember the Chicago Marathon being delayed or cancelled a couple of years ago for this reason.
Your core temperature cannot rise too much for many reasons, and your body will try to prevent you from doing so. Specifically,biological enzymes, being proteins, have a very specific temperature range in which they function properly. Also, the oxygen/hemoglobin dissociation curve becomes less favorable (shifts to the right) with a rise in blood temperature. Basically, you transport oxygen less efficiently. This isn't drastic, but is significant. Of course, higher heat also leads to more dehydration, which is in itself a major problem.