Every day, while you're watching television or driving your car or reading a book, you are analyzing what is in front of you. While that analysis may include complex internal discussions or just passing thoughts, your brain is endlessly analyzing your world.
Essays are short, informative pieces of writing that focus on a specific, chosen topic that comes in four traditional forms. An expository essay requires no real written analysis, just a presentation of facts or positions on that topic. Descriptive essays only require that a writer paint a picture using words that puts a reader inside the scene as if they're experiencing it themselves. A narrative essay tells a story of a personal experience, and an analysis of what the writer learned from that experience may be included. A persuasive essay requires that a writer make arguments that support their point and includes the writer's analysis of those points to reach a conclusion.
An analysis requires an examination that deconstructs something in order to draw conclusions and make decisions. For instance, your brain analyzes the signs and traffic at a four-way stop to determine how to proceed when driving your car. Watching a television show requires your brain to analyze what you are seeing and reach a conclusion about it by making references to what you already know about the story. In both instances, even without thinking too hard about it, you are taking apart the information or experience in front of you to reach a conclusion.
A written analysis requires a writer to look at many different pieces of information available on a topic to form conclusions about it. The writer's thoughts on a topic are broken down into separate points to explain their reasoning and the conclusion is based on the synthesis of all of those separate points to explain how they work together to prove a writer's main point.
The difference is simply in the definition. An analysis can be performed and expressed verbally, through the decisions you make, or through writing. Analyzing information requires breaking down a topic to see how it works and then drawing a conclusion about it. Not all the essays you write will include an analysis.
An essay, on the other hand, will always require writing. Writing an essay may include forming an analysis of the information in front of you, but it may also require that you simply compile the information in order to present it in an organized structure.