When you get an exemption, you don't have to do something. In the sixties, some people got exemptions from military service during the Vietnam War.
We don't often think of being left out as a good thing, but when you use the word exemption, you mean that someone has been excluded in a positive way. Children are exemptions, or deductions, on tax forms; the more children you have the less taxes you pay. Some non-profits are tax-exempt; their exemption means they pay no taxes at all. Exemptions also spare people from fighting in wars and doing some jobs. An exemption gets you off the hook.