The two men have not attempted to disguise their animosity or their professional differences on how to improve results in schools.
However, Professor Brighouse, director of education in Birmingham, insists his decision to leave the standards task force has nothing to do with Mr Woodhead.
He stresses that he is resigning in order to devote all his energies to Birmingham schools.
The appointment of Mr Woodhead and Professor Brighouse as joint chairs of the task force in 1996 was variously interpreted as a masterstroke in bringing together opposing forces in education or a cynical acceptance that the chief inspector would only participate if he were given a leading role.
Any hopes of a constructive relationship were dashed following the publication of the report on Birmingham's education services by Chris Woodhead's Office for Standards in Education.
Early drafts of the report implicitly criticised Professor Brighouse's style, and while these were removed, the final draft continued to query whether standards should have improved faster.
Professor Brighouse is thought to have put his reasons for resigning to David Blunkett, the Education Secretary, last Friday. While he is resisting pressure to involve the chief inspector, it is understood there have been recent exchanges between the two men over issues to do with Birmingham.
The resignation of Professor Brighouse is likely to cast doubt on the future of the task force.
However, the most serious fall-out is the recognition that there is a divide that cannot be bridged among education professionals. The Brighouse camp believes schools need to be nurtured and that "naming and shaming" is not helpful.