Why do dogs pant instead of sweating?

Because they have almost no sweat glands in the skin - only a few in the pads of the feet, which is why a nervous dog leaves damp prints on a floor. Fur would stop skin sweat evaporating anyway, so cooling has to happen somewhere that air moves quickly, and that is the airway.

Panting is a rapid shallow breath, several hundred a minute, moving air over the wet surfaces of the tongue, mouth and nasal passages so that water evaporates and takes heat with it. It is efficient enough for most conditions and it fails badly in two: high humidity, where evaporation slows, and any situation where the dog cannot get rid of the warm air, which is why a car in the sun kills dogs so quickly.

Flat-faced breeds have far less airway to evaporate from, and that is a welfare problem rather than a quirk. Horses and humans are the unusual ones - both sweat copiously over the whole body, which is a large part of why humans can run down faster animals over distance in the heat, and why a horse can be ridden all day.