The police decided not to prosecute. They decided to let it pass (maybe because it was a minor offence and not worth the time and resources needed to prosecute).
Is it exactly the same to say: the police didn't decide to prosecute. No. In this case, maybe someone else decided it for them (the attorney, for instance), but they did prosecute in the end.
the police didn't prosecute.
At school you learn how to negate with "didn't". Somehow, nobody at school teaches the negation with "not" (my own experience; don't feel addressed).
When do we use or prefer "not" over "didn't"?
Student or Learner