Hello, I will have to write an essay in class next week about the differences and common points between Annie John and The Crucible through the use of themes, quotes, and so on. I do not want you to do my essay, just giving me ideas please and some examples if it is possible. It is not something I can ask my parents because they do not speak English (they are French), unlike the parents of my classmates who are from United States, Australia, and Great Britain. (I am in an international section in a high school in France, even though I am not bilingual.) Thank you

There are far more differences between these two works than there are similarities. The Crucible is a play written by Arthur Miller, a white American playwright, in 1953. Annie John is a novel which was written over 30 years later in 1985 by Jamaica Kincaid, an Antiguan American woman. The authors are from different generations, as Miller was born in 1915 and Kincaid was born in 1949. 
Miller's play is written for performance. Thus, we learn about the events and characters in it from what Aristotle calls "mimesis" or imitation. Actors pretending to be the characters move around the stage, speak, and interact with each other. We have no special insight into their ideas and feelings other than what we infer from their words and actions. Kincaid's novel communicates in a mode that Aristotle termed "diegesis" or narrative. In other words, rather than seeing actors perform the characters of a novel, we have narrators tell us about their thoughts, speech, and actions. Annie John is narrated in the first person, meaning that the world we encounter in the novel is refracted through her experience and point of view and that we have access to what goes on inside her head.
Miller's play is historical, set during the Salem Witch Trials (1692-1693) some two centuries before Miller's birth in 1915. Its plot is based on Miller's research into an historical event. Its critique of the McCarthy era activities of the House Un-American Activities Committee (HUAC), is accomplished indirectly. The setting of The Crucible is a small town in New England, and the characters are ethnically uniform—with the singular exception of the slave girl, Tituba. Kincaid's novel is contemporary and set in the actual world in which she lives. In some ways, it is closely linked to her own personal experiences growing up in Antigua and then moving to New York. Although Annie leaves for England rather than the United States, the setting still reflects a globalized, postimperial, cosmopolitan world. 
The protagonist of Miller's play is John Proctor, a prosperous farmer, and the protagonist of Kincaid's novel is a young woman. One important parallel you can draw, though, is that both works explore the emotional turmoil of young women (Abagail Williams and Mary Warren in Miller's play) growing up and learning how to negotiate strong interpersonal emotions and emotional turmoil. In Miller's character of Tituba and the conjure women of Kincaid, folk medicine steps in where more scientific, European medicine fails to help with what are essentially psychological ailments. The contrast between male scientific and female folk medicine is especially important.