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Hip Hop is the whole culture, which includes Rap, beatbox, deejaying (especially scratching and mixing), etc etc.

Rap is just that...rap. Rhythmic verse in a lyrical style.

Rap is to Hip Hop what Heavy Metal is to Guitar Music. Just one branch or element of the whole.

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deejaying (especially scratching and mixing)
For the purist, this is called 'turntablism'. Proper hip-hop DJs who scratch are called 'turntablists'. Whereas the modern mainstream 'pop-hop' DJs just put a backing track on for the singing rappers or rapping singers to sing / rap over. And that type of thing is rap with a silent 'C'.