Learn English – “severe” as a verb: is it acceptable usage

They mean the same thing, but the second one calls attention to the fact that the exemption came from somewhere or someone.

Imagine you're talking to some bureaucrat at a counter. She smiles at your children, passes your money back, and says "You don't need to pay this; children are exempt."

Different situation. You don't have enough money, and children are not normally exempt. "Okay, we'll make an exception." She scribbles her name on a piece of paper and writes "Children are exempted from the charges by my authority."

You could still used exempted in the first case, but you cannot use exempt in the second one.

Accidently has an entry in the (full) OED, but it's labelled as non-standard:

Now nonstandard. In an accidental or chance manner; accidentally, by chance (contextually: by mischance). (In modern use only as an occasional spelling variant of accidentally adv.)

Interestingly, the first citation for accidently is (tentatively) dated 1425, only a few decades after the first appearance of accidentally (1398).

Merriam-Webster also includes accidently, which it accepts as a variant spelling:

— accidentally or less commonly accidently adverb