Basically, 'rap' is the spoken words over a backing of bass drums and keybaords, and 'hip-hop' is the actual music itself, so effectively they are parts of the same musical form, although the overall genre is covered by the generic name of 'rap'.
well rap is an actual style or singing where as hip hop is the whole culture of that genre of music. hip hop also includes breakdancing battle rhyming and graffitti but rap is the most popular feature!!!