A dog's skin is quite different, which is why you have never seen a dog with sweaty underarms. Most of the dog's sweat glands are located around its footpads. That is why, when a dog is overheated, you will sometimes see a trail of wet footprints that he has left behind as he walked across the floor.
Rather than relying upon sweat, the principal mechanism that a dog uses to cool himself involves panting with his mouth open. This allows the moisture on his tongue to evaporate, and the heavy breathing also allows the moist lining of their lungs to serve as a surface from which moisture can evaporate. In this way, the dog can manage a significant cooling of his body temperature.
Another mechanism that dogs use to try to cool off involves dilating or expanding blood vessels in their face and ears. If it is not too hot outside, this helps to cool the dog's blood by causing it to flow closer to the surface of the skin. This mechanism works best if the overheating is due to exercise, rather than a high outside temperature.
Cats do have some sweat glands, but their skin is covered in fur, so this minimizes the amount of cooling the sweat can provide. The paw pads have the most sweat glands. You may see damp footprints from your cat walking on a hard surface in the summertime.
Cat Owner's Veterinary Handbook explains that felines' sweat glands are found only in footpads. If the cat becomes overheated (or frightened), she secretes sweat through her paws. At this point, she may resort to cooling herself by panting or licking her fur. However, unlike with dogs, a cat's panting is usually more stress-related than heat-related.
If you feel that your kitty's panting is from overheating during those hot summer months, take measures to cool her immediately. You might want to clip her fur, but do not clip it too close to the skin, as this will increase her chances of getting sunburn.