'vacate the premises' is more formal and the term more reserved when it is some form of legal or official request for someone to depart. The police will often use this term, for example to get people out of a building. Also if someone is formally asked by staff to leave a hotel or club for misbehaviour or drunkenness, 'vacate' will be heard as much as 'leave' in the request.

"Vacate the premises" is extremely formal. A common use would be in a legal dispute. For example, if the rent is late, the landowner will send a demand to pay within three days or to vacate the premises. The Collins online dictionary lists "vacate" as among the 30,000 most-common words (thus not something that children or teenagers would know). https://www.collinsdictionary.com/dictionary/english/vacate The Collins online dictionary lists "premises" as among the 4,000 most-common words (thus children would not know it, but some teenagers would). https://www.collinsdictionary.com/dictionary/english/premises Additionally, "advise" is formal. The Collins online dictionary lists "advise" as among the 10,000 most-common words (thus children would not know it, but some teenagers would).

All of these are correct but I would say 'leave THE school immediately'. Using 'vacate the premises immediately' is basically asking the same thing but more formal and perhaps even more professional. I would be hesitant to use 'vacate the premises immediately' if they're young students as they mightn't understand or may feel more threatened.