The usage of slander, defame, vilify, and calumniate
Solution 1:
You defame by publishing information hurting good reputation - possibly embarrassing, claiming incompetence or negligence.
You vilify if you defame by presenting the person's actions to be evil. Posting photos of someone sleeping under a tree, drunk may be defaming. Photo of that person raising a hand on a child is vilifying. Also, presenting normally mostly benign actions, making them to appear as evil is called vilifying (e.g. claim a woman who wears a red dress is "dressing like a whore")
Slander is a defamation that is false. (note for a long time in England one could win a lawsuit for defamation even if the defaming claim was proven true. Slander is always a defamation which is a lie)
I'm sorry but I won't give you any details on calumniate, I can only guess that's not a very frequently used synonym for slander. I think it implies verbal (or textual) form of slander, which on itself may take other forms (performing, parodying) but don't quote me on that.
You should also note libel (per StoneyB's comment) which is the same a slander except committed in a fixed/permanent form - slander is always in a volatile form (speech, gestures), while libel is fixed - in writing, as a picture, video, voice recording etc.
Solution 2:
In some contexts, these words are interchangeable, but each has a different shade of meaning.
Calumniate means to falsely or maliciously accuse, especially of a crime. Defame and slander mean to spread tales about a person so as to damage their reputation, but defame refers more to the intended result while slander refers more to the action taken. Vilify means to make vile, generally by slandering.
See OneLook.com for links to many general reference works containing definitions and examples of usage.
Solution 3:
Slander is to defame by telling a lie. Calumny is to defame by telling what I know to be true with the intention of sullying someone's good name. In the case of calumny,the information is confidential and although it may reveal illegal or unethical behavior performed in the past, guilt or innocence has been resolved in some way or the revelation of it may produce more harm than good.