With a teacher account, you get:
With an Admin account, you get:
Not necessarily. If you purchase an Admin account, you only need to purchase accounts for the maximum number of students who might be assigned to use a program during their usage time-period.
For example, if you think you will have 100 students using the program in the Fall, 75 in the Winter, and 50 in the Summer, then if you have a Admin account, you only need to purchase 100 seats.