TSE-A is for the application of graduate study, if you are going to use TSE score for applying the graduate school, you need to take TSE-A.
However, if you are going to work as a professional, such as nurse, pharmacist, vet or physical therapist, you need to take TSE-P because these boards only accept TSE-P
Both TSE-A and TSE-P take exact same test, but I was told that ETS grade TSE-P more strictly. Therefore, it is easier for students to get TSE-A 50, but not professionals to get TSE-P 50.