What is the difference between conciliate and appease ?Feel free to just provide example sentences.

The difference is a bit subtle. Appease means to calm someone who is angry and demanding. That's a bit of a simple explanation, but it will work. Conciliate can mean the same thing, but is a little more broad in scope. Firstly, I should mention conciliate is a word you will probably never hear, even appease is a little "high-brow" for most people, but conciliate means to...you know what, I think it is example sentence time:

"The angry, pitchfork-wielding mob demanded they burn the witch. He had no choice, but to appease them." (he burned the witch)
"He lost in the final round of the gameshow, but they gave him a great concellation prize."

Here you can see, in my second example, it used in another form to describe a gift given to someone so they don't go away mad. You are trying to win back a little bit of their friendship and goodwill through pleasant behavior (gift giving). You do not have to "appease" them, because they probably are not angry and demanding (probably), but they might feel bad.