Sweating the vegetables makes them more tender in addition to drawing their flavors out into the rest of the dish. I would never skip the sweating process when following a specific recipe, otherwise you will wind up with hard bits of stuff in your final dish. For example, some recipes for a home made tomato sauce call for first sauteing some diced carrot, onion, garlic and celery first, which becomes a base for your sauce. If you were to skip this step, then just pour in the canned tomatoes, your sauce would have all sorts of hard carrots and celery bits in it. Whereas if you had sweated them down initially, they would be soft and blend into the sauce giving it it's more desirable smooth texture.
Sweating is just basically sauteing but without letting the vegetables get much color. In other words, it would be like sauteing some diced onions, but don't let them get to the point of caramelizing and browning, which will change their flavor.