Moliere's original play had a Roman Catholic priest as the character Tartuffe. But, because the farcical plot centers around some rather outrageous religious hypocrisy, and because France during Moliere's time was a Catholic country in which many clergymen had political power, the play was banned for its use of physical humor and ridiculing of a Catholic clergyman. After five years, Moliere revised the character to that of a Puritan. He was then allowed to put on performances.
Another very controversial aspect of Moliere's Tartuffe is the subtle suggestion that the monarchy (represented by Orgon) can be deceived by the clergy/religion (Tartuffe) and can lead its citizenry to demise. This is, indeed, a dangerous theme to suggest and one that certainly brought about the censure of Moliere's play.
That Orgon is so deluded about Tartuffe that he must be made to hide under the table to overhear Tartuffe's proposals to Elmire is insult enough, but that Orgon is so crass that he does not come out until Tartuffe nearly assaults Elmire is an assault to both the ruling class and to the clergy.
Answering the actual question you first posted, the specifics of Tarfuffe as a controversial play are that Moliere openly exposed the quiet social preoccupation of the time, which was that so-called moralistic and religious people were using the church as a conduit to obtain money, status, and free things from followers.
The play is not objectionable under the worldview of the Enlightenment because that was precisely the route that the 17th century adopted: To move away from the Renaissance and the abuses of the clergy. However, it may have been controversial to make a man thought of as a saint in a scene trying to seduce another man's wife.
I can answer the first part of your question which should help you with second part. It would be helpful for you to refer to eNotes' historical context and introduction to help with a better overall understanding of this play, which may also assist you in at least part of the rest of your questions.
First, Tartuffe is hysterically funny; if you ever have the chance to see it on stage, go for it. It is absolutely wonderful. By the way, it is a farce (a satirical comedy).
The play is about a man (Tartuffe) who tries to ingratiate himself with a wealthy family in order to trick Orgon (wealthy man of the house). To get an "in" with Orgon, Tartuffe pretends to be extremely pious (holy). Orgon takes him at face value, seeing a saint, while never looking beneath Tartuffe's facade; he treats his house guest better than his family--even as far as to plan to marry his daughter to this relative stranger.
As with life, when Orgon's friends warn him that Tartuffe is not what he seems, Orgon is too close to the situation to see Tartuffe for the hypocrite and liar that he is. When Orgon overhears Tartuffe's attempted seduction of his wife, his eyes are finally opened. In the end, after Tartuffe shows his true colors and almost gets away with the bank, the family is delivered from the clutches of this scheming con man.
With all this in mind, remember this play is written under the "patronage" of King Louis XIV of France, who (like Elizabeth I in England) was a true supporter of the "cultural arts," which thrived during his reign. However, the country was still very much "under the thumb" of the Catholic church, with no separation of Church and State.
Two reasons Moliere's contemporaries would be displeased with the play are, first, the objectionable character of Tartuffe who uses all things holy and sacred to pretend to be a truly God-loving man, when he is, in fact, a schemer beneath his "disguise." The Church actually banned the play--it did not see the stage for five years; it was only after Moliere's appeals to the King that it was finally performed without obstruction.
The second reason, simply put, is that no one likes to think that his own behavior is being exposed to ridicule. Those like the character of Tartuffe ("church angels, house devils") would take it personally that Moliere would draw attention to their disgraceful, hypocritical behavior which was counter to the teachings of the Church.
This play is a satire, a piece of literature that uses humor--in this case harsh humor--to make a point about something within society that the author brings to ridicule. Tartuffe's hypocrisy a major theme in the story, as is Orgon's foolish gullibility in that he cannot see Tartuffe for the fraud he is, refusing the counsel of friends, and placing the wishes of his guest over those of his own family.
When it comes to finger-pointing, Moliere wasn't letting anyone off the hook easily. His finger wagging would have made the hypocrites within his society (and there are always many in any society) extremely uncomfortable. This did not, however, stop theater-goers from attending the play. Candid it may be, but it is also very entertaining: perhaps even more so to those of us who can, at least secretly, identify with perhaps both of the main characters-- saying one thing and doing the other, and/or being too close emotionally to a situation to see it for what it really is.