Cats rarely engage in exhausting exercise for as long as dogs do, so they don't very often run out of breath. That's why you seldom see them panting. Dogs evolved to be pack hunters, which requires them to have the stamina to chase prey long enough to simply exhaust it, as well as themselves. Cats evolved to be solitary hunters, which requires them to be quick, stealthy, and smart, and catch the prey before it gets a good head start and before either party runs out of breath.
Dogs and cats both play according to their hunting instincts. Dogs run around in packs until they fall down panting. Cats chase each other around the room once and then they take a six hour nap.
Cats and dogs have much different shaped skulls and nasal passages, which means that colds and other respiratory ailments give them different visible symptoms. American Persian cats, which have lately been bred to have those ridiculously flat faces (as opposed to the British standard, which is more old-fashioned) commonly have sinus trouble. They sneeze and wheeze and breathe through their mouths and need daily doses of antihistamines just like people.
As for the different ways that dogs and cats behave in similar situations, you have to remember that possibly the most significant difference between the two otherwise rather closely related carnivores is that dogs are social animals and cats are not. We think our pet cats are naturally social creatures because they seem to behave that way, but it's actually a strange phenomenon called "neoteny" that only occurs in captivity. Something in our relationship with our cats (and quite possibly in eight thousand years of domestic selective breeding) makes them revert to the behavior of kittens when they're around us and have all of their basic needs fulfilled. But still, they are instinctively solitary creatures as adults. They don't have the complex behavioral instincts that dogs and other pack animals do that is required in order for packs to function.
Cats may or may not have a sense of dignity. It's quite possible that a creature that leads a primarily solitary existence would still care how it looks to others of its kind who might be thinking of encroaching on its personal hunting territory, to a potential mate, and to other predators.
Dogs have a whole lot more to think about. They've got their pack hierarchy, which is as complicated as a bunch of humans preening themselves to get ready for a night club. Or for a gang fight, which actually has a lot more in common with what our pack of hard-headed Lhasa Apsos do for fun and exercise.
