Cushioning is the property of the shoe to absorb part of the impact. Not all running shoes have the same amount of cushioning. Maximal running shoes usually have thicker soles.
The correct terminology is stack height, and they provide more shock absorption, and they also weigh more. Aside from personal preferences, maximally cushioned shoes are recommended to heavier runners, and in general, for day training, especially for the long-distance runs where the most important thing is comfort and injury prevention.
Minimally cushioned running shoes are lower to the ground, weigh less, and provide less protection from the impacts. Usually, they’re reserved for lighter, more efficient runners, who use them for faster training days or even races.