If you're the subject of an essay, either you or someone else found you interesting enough to write about. If you're the subject of a case study, you're probably seeing a social worker or doctor who finds you interesting enough that other people in their field would like to study you.
An essay is a short, informative piece of writing with an introduction, a body, and a conclusion. Depending on the type of essay, the purpose is to tell a story, argue a point, describe a scene or event, or simply to present information. Essay topics can vary from history to current events, from science to philosophy, and from a personal experience of surviving a natural disaster to describing what you did over summer vacation.
A case study presents the information gathered by a social worker or doctor on a person or group of people within a larger context of study. For instance, a case study of a person may be used to guide social work policy in consideration of the effects of current policy on that individual. A group case study may be used to help doctors decide how to best treat patients who have survived traumatic events or who suffer from a particular mental illness. Case studies are generally used as examples of larger phenomena, such as how several different individuals experienced the same effects after similar forms of abuse.
In short, a case study is used by scientists to form opinions and propose approaches about larger issues.
To research information for writing an essay, a writer can look in their local library, on the internet, in past writings about the topic at hand, or by performing interviews with people who have knowledge of the subject.
For a case study, the writer needs a human subject or group of human subjects. The writer of a case study must record in-depth information on the details around the person or group of people studied, disclose the methods used to gather that information, and follow strict ethical guidelines around studying human subjects.
While both essays and case studies are pieces of writing meant to inform others, an essay can inform anyone on a topic, while the audience for a case study will be limited to social or life science colleagues or researchers in those fields.