First of all, I agree with everything @shog9 said. To add just a little more:
I used to have a customer in the wholesale gourmet food business. I was their IT department, but also their unofficial kosher consultant. Absolute best lox (Samaki), some really great chocolate. Ah, those were the days. Among other things, they sold a ton (literally - this was wholesale) of sea salt. They basically admitted - all salt is almost the same. But they sold plenty anyway.
The shape/texture difference can be really important when sprinkled on a salad or another dish right before serving. Cook or bake with it, and once it dissolves it is no different from regular salt.
As far as other chemicals, they can make a difference in taste. I wouldn't count on any health benefits as those other chemicals are not in any controlled formula/mixture. While there are some sources I found indicating as much as 10% of sea salt being "other stuff" besides sodium chloride, most sources show 2% or less. A difference is often touted about sea salt having less sodium than regular salt, but that is almost always per teaspoon or other measurement of volume. Sea salt has less sodium per teaspoon than regular salt because of the large, irregularly shaped crystals which mean there is less everything per teaspoon, not just sodium.
One other thing to watch out for is iodine, or lack thereof. Iodized salt is a thing because otherwise many people would not get enough iodine in their diet. Non-iodized salt is available for those who need it (and for Passover certification). If you replace all your regular iodized salt with sea salt then this may be a concern. If you just replace some of your salt, but still use iodized salt for cooking and baking then this is less likely to be a problem.
Bottom line: If you like the taste or texture of sea salt, sprinkle it on your food and enjoy. But don't waste money cooking with it.