If the content of the test is concerned, then the test dates & even the content of the test, the day test is conducted by IDP & British Council is same.
Test Dates are scheduled by Cambridge & are same for both IDP & British Council.
If you appear either under IDP or British Council, the test paper, the questions, everything is same.
So, Reading / Listening / Writing is exactly the same with entirely same questions.
So, Reading & Listening will also have same answers & will be checked with pre-defined answers, so doesnt makes any diff whether given from IDP or British Council.
Also, Writing & Speaking, one can tell oneself how it went, + / - 0.5, nothing more than that.