Does kosher salt has a different structure than regular table salt or it is an issue of crystals size?
Why some recipes want us to use one instead of another?
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There is no chemical difference between table salt and kosher salt - both are sodium chloride. Kosher salt just has bigger crystals. However, some table salt brands are iodized, meaning that iodide salts have been added to help us maintain a healthy dietary intake of the essential nutrient iodine. Kosher salt is never iodized.
Sea salt is a different matter. Since sea salt is obtained from sea water, it may have other minerals in it, which perhaps alters the taste.
Kosher salt is used in some recipes as a matter of taste - the bigger salt granules take longer do dissolve, and so there may still be little pockets of salty crunchiness. Kosher salt is also used when this property is necessary, for example in the curing of some meats. For more on the differences in culinary use between the various types of salt, you should consult Seasoned Advice.
All types of salts with names like Kosher salts, Himalayan pink salt, table salt, sea salt or iodized salt are mainly sodium chloride (NaCl). There might be trace element differences hence a difference in taste.
The key information is missing from the accepted answer as well as the Wikipedia link. Kosher does not mean that the salt crystals are larger. Kosher is just a religious label which shows that the food item has been prepared in conformity with the Old Testament's laws. The label "K" or Kosher of food products indicates that this salt does not contain any porcine product, nor it has come in contact with them and other additives which do not conform to biblical laws. Kosher is almost similar to the label "Halal" (=permitted according to Muslim laws) on many food and cosmetic products.