What nutrients do I lose in the water when I boil vegetables?
Vitamin C and the B complex vitamins are water soluble.   Thiamine and folate are also heat labile. 

Overall there are hundreds or often thousands of kinds of small molecules (not protein or DNA) in the cell and many of them are water soluble or heat labile.  Plants especially have many unusual small molecules.  Its not known how many of them affect human health, but may affect factors like the gut microbiome quite a bit and have an effect on well being if not mortality. 

Some people think that live uncooked vegetables have enzymes and other proteins which are good for you since cooked proteins are denatured and may be taken in differently.  At the very least they are probably easier to digest when they are not heat denatured.