Pure Finance (MSF) majors tend to be quantitative analysis, so the requirements for your math scores will be higher. Of course, some management courses, such as financial management and enterprise strategy, will also be slightly mixed in MSF courses. Among the top 100 schools in the United States, less than 50 schools offer MSF.
In the direction of finance under MBA, this major is more inclined to the curriculum of management, and the courses that really involve finance are not too deep, so they are more extensive than MSF.
Financial engineering is an interdisciplinary subject connecting finance, mathematics and computer science, and it is also divided into four aspects: financial engineering, quantitative finance, financial mathematics and computer finance. Its courses are jointly taught by the business school, mathematics department and engineering college of the university, which is more difficult to learn and apply for.