First, I'm not an expert, but rather a well-informed consumer. That being said...
1- +1 re: don't trust the calorie counters on gym equipment
2- Most of Danno's repsonse doesn't make sense to me. The only way I can make sense of it is if he is assuming one is riding a bicycle for a longer duration that one would be running. In that case, one may not feel like he/she is working as hard riding the bicycle as when running for a shorter duration, though the actual caloric expenditure may be the same. Either way, working harder = greater calories burned, though your perceived effort may be skewed by which type of exercise is more desireable.
Sweating is a function of heating up- all things being equal (ambient temp., humidity, air flow, time spent exercising, etc.) the hotter one gets the more one will sweat to cool down. Remember, evaporation causes cooling. All energy is heat energy, including stored fat - it's simply heat energy. Therefore, the hotter one gets (ie, the more one sweats) the more heat energy is being released. This is why one gets hotter when exercising. Incindentally, this is the very reason why the noted physiologist Dr. Ellington Darden advocates drinking ice water: it takes approx. 7.6 calories to metabolize one cup (8 oz.) of ice water because it has to be heated up to body temp. before it can be digested. If one drank a half-gallon of ice water per day and did nothing else differently, one would lose c. 6.3 lbs in one year.
There are many theories on which kind of exercise lends itself to increased fat burn. The most prominent are: 1- lifting weights increases active tissue (muscle) which burns more calories in the long run 2- long, slow exercise session burns fat (9 calories per gram) instead of glycogen stores (4 calories per gram) 3- increasing one's base metabolic rate (BMR) through high-intensity interval training (HIIT) such as interval training and sprints. It's my understanding that most serious athletes train with a mixture of all 3.