An intact garlic clove has almost no smell. Its sharp aroma and flavor emerge when its cells are broken and two substances combine to produce a new compound. The more cells are broken, the more of this compound is produced. That is why whole garlic has the mildest result and crushed garlic the sharpest, with sliced and minced garlic between those two extremes.
Garlic cells contain two things that are separate from each other: an odorless precursor and an enzyme that transforms it. When the clove is intact, these two substances never meet, which is why unpeeled garlic does not smell. When a knife or press breaks the cell walls, the two come together and the sharp compound that gives garlic its character is produced at that moment. In other words, the pungency is not already waiting inside the garlic. For the same reason, a peeled but otherwise intact clove also remains odorless for a long time. That is why a whole head of garlic gives off no smell when held in your hand, but crushing a single clove makes the kitchen smell of garlic.
This leads to a clear ranking. Garlic left whole or roasted in its skin undergoes the least cell damage; the result is a sweet, mild, almost creamy flavor. Thickly sliced garlic has a moderate pungency. Finely minced garlic is noticeably stronger. Crushed or pressed garlic gives the sharpest result because nearly all the cells have been broken. Crushing it with the flat side of a knife and then mincing it is a middle ground and is preferred in many recipes.
There is one more detail: the reaction takes time to finish. If crushed garlic goes straight into the pan, heat damages the enzyme and some of the compound never forms. If you let it rest for a few minutes after crushing, the reaction finishes and the flavor becomes fuller. This rest also helps when using it raw; garlic added to a yogurt sauce tastes more settled if it rests for a few minutes. Garlic pounded with salt in a mortar works the same way; the salt makes it easier to crush and rounds out the flavor.
Cooking time also changes the result. Garlic added to oil at the beginning of cooking is exposed to heat for a long time; most of its pungency disappears and a sweet base remains. Garlic added at the end keeps its pungency and stands out. The two forms can both be used in the same recipe: some at the beginning and some at the end. Garlic burns easily as it sautés in oil, and burnt garlic tastes bitter. So add it to the pan after the onion and turn the heat down somewhat at that point.
The choice depends on the dish. If you want the garlic to stand out, crush it and add it late; if you want it to create a background base, slice it and add it early. In raw preparations such as garlic yogurt, cucumber-yogurt sauce, and other sauces, use a measured amount because no heat will reduce its pungency. If the sprout in the center of a garlic clove has turned green, remove it; that part tastes more bitter. Fresh garlic is noticeably milder than dried garlic and can be used more generously. If garlic odor lingers on your hands, rubbing them against a stainless steel surface can noticeably reduce it.