Scientifically speaking, why are processed sugars and salts so much
worse for us than natural sugars and unprocessed sea salt? Have
studies been done to prove this?

Scientifically, sugar and salt are sugar and salt.  Whether you
process them or not, it's still NaCl and sucrose(table sugar),
fructose(fruit sugar). Artificial ingredients are another matter.

Now, un-processed foods eg. sugar may contain a variety of impurities
that may have some nice flavors, antioxidant qualities, minerals that
processing removes in order to purify.  Some processed foods(like
iodized salt, vitamin fortified milk, flour etc) are made to replace
some of the key nutritional value that is lost in processing.

As far as studies go, I would submit that none of them would be valid
in light of the fact that as long as you're eating a balanced diet,
you'll not be able to tell where you got your vitamins and minerals. 
I don't think it's known whether or not you can survive on a diet of
froot loops, power bars and steakum, but I would think most
nutritionists would call you an idiot for trying.

Chugs.

Actually, the are not. One could argue both ways.
Natural sugars and salts are not treated by gases in order to kill any
harmful bacteria or parasites that may be growing on them. "Organics"
they are called. Organics are washed but never gased. Gasing destroys
some of the nutritional value of the food and can leave residues that
have not been studied enough to now weather or not they can case
certain types' of cancers. Sugars primary element is magnesium. Salt
is iodized. But, Salt already has iodine in it from the sea, Sea Salt.
So you must make a choice, you can take your chances and potentially
contract parasites or bacteria from ungased food or you can contract a
possible form of cancer. These facts concerning cancer have yet to be
proven.
Stay Healthy!
Spinnster