Sea salt usually comes with a lot more minerals, and sometimes even remnants of seaweed. (japanese sea salt) and so has a little more character. This character doesn't come through as much in a longer cooking process like a soup or stew, but is better savored sprinkled over sliced steak or maybe to finish a piece of fish.
Some people suggest it's saltier, but I don't know if it actually is. A big part of what can affect salt's "saltiness" is the granule size - If you imagine difference between filling a bulldozer scoop with boulders or with gravel, the difference is considerable.
This holds true on a smaller scale, so there is a huge difference between a tablespoon of kosher, table and fine sea salt. For this reason a lot of recipes call for weights of salt, or at least specify the type of salt to be measured.
