Charlotte Bronte was raised partially by her religious aunt after the death of her mother.  Jane Eyre was also raised by her religious aunt.  Charlotte and her sisters were sent to a religious school.  While at the school, two of her sisters, Elizabeth and Maria, became sick and died.  In the novel, Jane was sent away to a religious school.  While there, she befriended a fellow student, Helen.  Soon Helen became ill and died.  

Charlotte left home to become a governess when she became an adult.  In the novel, Jane also became a governess as a grown woman.  Charlotte met a minister who gave her an offer of marriage.  She was not in love with him, and after consideration she refused his proposal.  Jane met St. John Rivers, who was also a minister.  When he proposed, she refused.  

In the end, Charlotte did marry the minister.  In the novel, Jane married Mr. Rochester.

What are the aspects of similarity between the author of Jane Eyre—Charlotte Bronte—and the heroine?

Several aspects of the heroine of Jane Eyre are autobiographical in nature. Perhaps the most obvious is that the heroine, like the author, is a young, unmarried woman who is socially a member of the gentility. In other words, she is from the upper classes of English society but not a member of the nobility. This means that most forms of gainful employment would be considered socially inappropriate for her class; being in "trade" or learning a craft was considered vulgar. Marriage was the typical road to economic security for women of this class, but neither the author nor Jane were conventionally physically attractive, limiting their marriage opportunities.

On the other hand, both author and heroine were in financially precarious circumstances, albeit for slightly different reasons. One of the few socially acceptable ways for young women in such positions to earn money was serving as a governess; both the author and Jane did so. Of course, at the end of the novel, Jane inherits wealth and marries, which did not happen to the author.

Both author and heroine were members of the Church of England and brought up within the evangelical wing of that church. The author's father was a clergyman and religious themes are important in the novel.