Both nouns refer to a small, exclusive group of people. Clique suggests a fairly closed group of people, often forming within a larger group, whose members have similar interests or status. Clique has a negative connotation, and suggests an association based on a sense of superiority to everyone else (a clique of popular kids) or on a desire to hold all the power: the ruling clique. Coterie can have positive or negative connotations. It can suggest a close-knit circle of friends, artists, scholars, or scientists: a literary coterie. Coterie can also suggest an exclusive set of people who associate with a prominent person, whose coterie it is: surrounded by her coterie of advisors; his loyal coterie. Here there’s more potential for elitism. As with ruling cliques, powerful coteries in government are usually undesirable, unelected holders of power.
Both words refer to setting something going or putting it in motion. Start has a wider range of usage. We can say someone starts a car, starts a club, or starts an argument. Instigate, which derives from a Latin root meaning “to prick” or “to goad,” refers to prompting a human activity by urging or spurring it on. The object of instigate is usually an act of aggression or hostility (instigated a riot; instigated a war), so the word tends to have a negative connotation. However, instigating change is one common and more positive use, and both instigating a project and instigating a discussion are also fairly common and less negative.
Both words refer to arriving at the final point or stage of something. When a movie ends, it’s over. The sense of end we’re concerned with here can stand alone, as in the example above, or it can be followed by a prepositional phrase stating how something ended: The scene ends with a fade-out. Culminate works as a strong synonym for end in cases like this because it’s normally followed by the preposition “in,” or sometimes “with.” Culminating with a fade-out could be a bit self-contradictory, however, because culminate, which comes from a Latin word for “top” or “peak,” suggests ending at a high, ultimate, or conclusive point: This nail-biting scene culminates in a shocking revelation.