Why do states and the federal government exercise oversight responsibility over not-for-profit, tax-exempt corporations? How do states and the federal government differ in the way they exercise this responsibility?

This is done inorder to prevent fraud and protect the interest of the public.

Explanation:

Not-for-profit corporation as the name suggest are establishment and companies that are into the running of various business interest with the sole aim of not profiting fro it (that is a noble cause). Some of the business interest which the not-for-profits corporation engages in include the runing of the hospitals, colleges, schools which is strictly meant for noble cause.

However, some people hides under this corporation to commit some financial infractions towards the government knowing fully well that they are exempted from tax payment. For example, paying money into the account of the not-for-profit organization from a questionable source and going ahead to spend such money in private business interest like personal house purchase rather than the noble cause such organization was set up for in the first place.

In summary, the government (federal or state) exercise oversight responsibility over the not-for-profit corporation inorder to prevent fraudlent transcations from being carried out in them while at the same time protecting the interest of the public for which such was established in the first place.