(Not a teacher)
There is little or no difference between the two options. Both tell us that you are a student and both "at" and "from" connect you to a given school.
If are away from your school and you want to tell someone which school you are from then "from" might be the better choice, but "at" still means the same thing.
If you say, "I am a student from X school, there is a very slight chance that you are not a student there, but have been visiting, doing research, etc. This is quite unlkely, but I prefer "at" because it unquestionably indicates which school you attend.
Student or Learner