A graduate school personal statement is for convincing admissions committee that an applicant is fit for an academic program and has something to contribute to the department. It can be in the form of a personal statement or statement of purpose. A personal statement discusses more you. A statement of purpose (SoP) shows your personality but also explains things that influenced your career path, professional interests, and plans.
The difference is that a SoP answers questions by the school in a concise way, but a personal statement has a story telling element that makes it more engaging.
A graduate school essay is one of these forms is a way to present information that portrays you as the right candidate for an academic program. Only a few members of the department faculty are available to read essays by thousands of applicants. You should make yours sharp as possible to convince the busy readers that you can be meaningful contributors to their department and college.