The EDL is a group which claims to protest extremist Islam but actually protests all Islam. It is run by a football hooligan who was arrested for leading football hooligans at Luton Town FC against Newport County FC last year and most EDL marches are on Luton Town away days for the convenience of its leader.
The BNP are a political party which stands in elections. It is probably fifth or sixth in the UK, discounting all the regional parties like the SNP and Plaid.
Their platform is racist, they were formed out of disaffected members of the National Front. Since Griffin took over he has attempted to change the image of the party, but not the substance, though he has seen that Islam has far more opportunities than Judaism to gain support so the BNP have been very silent on their anti-semetism and even claim to have a Jewish member, though she is married to a non-Jew and hates Muslims more than anything.
The problem with the new image change is that the old keeps getting in the way. Chris Hurst was caught by the Sun newspaper recently giving the Hitler salute and shouting "Sieg heil" at a concert for the favourite band of the Norway killer. He was "kicked out" of the party because he showed the BNP for what they are, but he will be back just as Dr Phil Edwards was caught on camera listening to a racist telephone call he had made, now Phil is writing for the BNP again.