It is a feature of minority parties on the extreme left and right that they schism and split frequently. As a party grows it gains members with a wider range of opinions. Often this leads to individuals and their followers believing that the leadership is making compromises and is insufficiently "pure". That individual sets up their own organisation, which grows and then the cycle begins again.
The BNP was founded as a splinter group within the National Front, and Britain First was a splinter group within the BNP. In terms of actual policies there is not much difference. Britain First portray themselves as defending Christians, against Islam, and the religious aspect of Britain first is absent from the BNP and the NF.
There is very little difference between the BNP and the NF. Both are neo-fascist or neo-nazi, white-nationalist, racist parties. The BNP was founded owing to personal differences between members of the leadership of the NF, not because of any ideological difference.