There is no difference between IDP and British Council IELTS tests. IELTS is owned by both British Council and IDP with one more partner - Cambridge English Language Assessment. Simply put, they all form one company for IELTS.
You need to note that both British Council and IDP conduct IELTS tests but don't compose them. The test papers or materials come from Cambridge English Language Assessment. In short, you get the same questions whether you book through IDP or British Council.