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Sweating is done at at lower temp and, usually in the case of onions (but not always), with the pan covered. The onions release quite a bit of their moisture content and essentially stew in it and soften. This promotes a fuller onion flavor than saute-till-soft cooking (though not of the intensity of caramelization) but does not reduce their volume much since evaporation is minimized.
Cooking, especially sweating, will dramatically soften the bite of any of the onion family==eliminate it, really--and is especially appropriate for garlic and leeks which can tend toward an
increased sharpness or bitterness if sauteed, paricularly if allowed to brown at high heat. (See
this recipe.) Leeks nearly always need to be sweated; garlic can be added to the pan at high heat, say, after the onions are caramelized, but should only be sauteed till fragrant in most cases--about 45 seconds.
One can saute onions to the soft/translucent stage for most recipes--sweating not required--or can saute further till lightly browned in spots or more. Care needs to be taken at this stage--i.e., frequent flipping or stirring is necessary--so as not to over-brown the spots that brown first while other spots are just taking on color. Over-browning can introduce sharp, bitter flavors to the finish.
When even caramelization is essential and/or when doing a mass of onions (in both cases, such as when making onion soup, onion 'jam', or onion confit), sweating first, covered, is a good idea. This way the onions release much moisture, as expected, but aong with that moisture some of the natural sugars are released. Then, when the onions are fully soft and they've released all they're going to (about 30-40 min along), the pot is uncovered and the heat raised a bit to evaporate off water. Stirring during this process evenly distributes the liquid as it reduces and brcomes syrupy, promoting even caramelization and a beautiful finish. (Many cooks add a little sugar when uncovering the pot to bosot sweetness some; the sugar dissolves well into the moisture and gets evenly distributed during evaporation and stirring.)
Two other times when sweating is helpful: when sauteing mushrooms (they are best sweated first--though the temps need not be very low--because the give off so much moisture), an opening sweat can pull out the moisture which then evaporates quickly when the heat is raised for browning; and when you have other things to attend to but need to soften other vegs that run the risk of bitterness if not attended to--like bell peppers (and other peppers), or that contain a lot of moisture that really needs to be evaporated off if some browning (and flavor development) is going to occur--like most summer squashes. Peppers can usually be sauteed till soft or till lightly browned here and there, with frequent pan flipping or stirring as the cook, but if you've meat to prep, a child requiring attention, whatever, sweating till you can get back to the stove can help.