See, we understand this as you know that IDP and British council are two organizations that conduct IELTS exams, there is not much difference between the two, almost both provide the same services.
But if I talk about my own experience, then I know more about IDP, not that I have no knowledge about the British council, just we can book our exam in both of them. It depends on you what you like.
If you know more about these two organizations, you can book your exam.
TRF (Test Report Form) will also be the same for both, just the name of the organization will be different.
So today you must have understood that we do not have much difference between IDP and the British Council, you can take whatever service you get in your coaching center.
By the way, the big institutes get registered with both organizations so that the students do not face any problems.