Since you were appointed to your post (and could be sacked from it) by the people who pay the bills to fund the police (i.e. the local authority), you can't be regarded as truly 'independent'. (i.e. in theory, your employers could put pressure on you not to 'rock the boat', as a member of the Police Authority, by hinting that your job prospects could be at risk).
Although (most) teachers are paid by local authorities, they are appointed by (and answerable to) the school's governors. Thus there is no conflict of interest.
Chris