An affront is an insult, indignity, or something offensive. As a verb, it means to insult or offend. Affront comes from comes from French affronter "to face, to brave, to confront". It can be used in a sentence as, "These laws are an affront to our free speech."
Aspersion comes from aspergere "to sprinkle on, spatter" based on
ad- "(up) to, on" and sparger "to strew, scatter." An aspersion
means a spattering or sprinkling, especially of holy water. It also
means that which bespatters or besmirches someone's character,
slander, defamation of character. People usually use the term 'cast
aspersions', as in spatter someone with metaphorical mud.
Incidentally, Asperger's syndrome is named for Hans Asperger and
totally unrelated to the word or word root above.