Unfortunately, this is a vague question, through no fault of your own. There are simply so many additives that each harm you in different ways. I will try to generalize though.
Most additives are linked to various cancers, weight gain, heart disease, diabetes, stroke, depression, sleep apnea, etc. The list really never ends.
Most people might shrug that off thinking everything is linked to all those conditions, but the reality is, most of the time those conditions occur from your diet, not other factors. They affect you throughout your life, not just when you are older. -- Wish I could that.
Take a simple additive like High Fructose Corn Syrup. Going into detail about HFCS is beyond the scope of this answer, but feel free to read at http://midsectionfat.org/what-is-sucrose/
Another excellent resource that really gets into all the different types of additives and diffrentiates the safe ones from the harmful ones: http://www.cspinet.org/reports/chemcuisine.htm