Is it healthy or more tasty?
As Kosher salt goes there is also a difference in the types. Many chefs (myself included) recommend and prefer Diamond Crystal brand Kosher salt. Diamond Crystal (owned by Cargill) uses a patented process of producing salt known as the Alberger Process. In in the interest of keeping the explanation simple, essentially it creates flat salt crystals with a hollow diamond shape and jagged edges. The hollow shape provides for quicker dissolution and the jagged edges help it stick to food better rather than bouncing off as is the nature of the cubic structure of basic table salt. For this reason many seasoning and food manufacturers use Alberger salt as a "dry emulsifier". I was on a tour with Paul Prudhomme at his seasoning plant near New Orleans and he explained that the jagged edges help to keep it suspended and blended with other seasonings and dry ingredients.
Morton's kosher salt is composed of large crystals that are rolled to flatten them. They are flat but not hollow so take a bit longer than Diamond Crystal to dissolve.
Morton's Kosher salt also has Yellow Prussiate of Soda added to it as an anti-caking agent ("when it rains it pours") which tends to leave a slight bitter (not as bad as iodine) taste in the back of the throat. Diamond Crystal brand is simply salt.
Alberger salt has a lighter bulk weight than granulated salt so 1 teaspoon of Diamond Crystal salt is going to be less salt than either table or Morton's kosher salt. Look at the two packages side by side in the store sometime. They are both 3# boxes but the Diamond Crystal box is much taller due to the increased volume of the hollow crystals. Looking at the nutritional information panels you'll also notice the serving size for each is 1/4 teaspoon but the Morton's has 418 gr. sodium per serving vs. about 200 for Diamond crystal (again, due to hollow crystals). Some people recommend increasing salt by 25% when using Diamond Crystal in recipes. In cooking you can easily salt to taste and know that since it dissolves more readily you should be able to determine seasoning by taste without oversalting. In baking I typically use it measure for measure and haven't had any major issues yet. In yeast doughs however you must be cautious to make sure that the crystals do dissolve in the liquid so as not to cut through the gluten strands during the kneading process (as well as to evenly disperse).
Additionally, regular salt contains iodine (added to most table salts since the 1920's) while Kosher salt does not have any added ingredients mixed into it....
There are no differences health-wise. They are both the same compound, in essentially the same form. The only difference is the shape of the crystals.
Chefs prefer it because it is easy to grip with fingers. Table salt is too fine, and runs between your fingers, making it difficult to get a consistent amount when adding a pinch.
The flat crystals work better on meat (the name comes from the process of 'koshering' meat, salting it to draw out the blood) because they dissolve more slowly and have more surface area per grain. Whereas table salt would create a spot of high salinity, kosher salt tends to dissolve and coat the meat evenly with salt.
Edit (based on comments):
Kosher salt does not contain iodine, which may present a health risk. A diet deficient in Iodine results in goiter, a condition which causes swelling of the thyroid gland.
In the past, persons mainly in the central United States (seafood is a good source of iodine) developed goiter in high numbers, and leading up to WW 1, many young men were turned away from service due to iodine deficiency. It was decided that since everyone ate salt, adding a small amount of iodine to common table salt would solve the problem. And so it has. Goiter has dropped significantly since salt started to be iodized in the 20s.
Kosher salt also may taste better due to the lack of iodine and anti-caking agents. Potassium iodide (usually used for iodized salt) has a bitter flavor, and the anti-caking agents may be unpleasant in texture (usually silicon dioxide, which doesn't have any flavor).
In addition, the shape of the salt crystal changes the perceived level of saltiness. Since the crystals tend to be larger, they dissolve more slowly, meaning you perceive less saltiness per gram salt consumed. It also has less salinity per volume due to the irregular shape of the crystals and their rough texture. This makes over salting less apparent.
I've long suspected it's merely a fad.
However, Jeffery Steingarten says he can taste the difference ...