Hip hop, as a movement, started in New York in the 70s with DJs and MCs like DJ Kool Herc, DJ Afrika Bambaataa, and members of the Zulu Nation. This movement incorporated various artistic media, including rapping/MCing, mixing and turntabling/DJing, break dancing, and graffiti art (some people also include beatboxing). Thus, to me, it seems comparing rap and hip hop straight up is somewhat difficult because, originally, rapping formed one of the pillars of the hip hop movement. I guess, in the end, the historical difference between rap and hip hop is that rap is one of the art forms popularized by the hip hop movement, while hip hop, in all, incorporated many different art forms.

Like /u/elev57 said, hip hop culture is the incorporation of different forms of media. Generally, the four elements are rapping(MCs), producing(DJs), break dancing, and graffiti.

The difference between the two is that originally, hip hop was a subculture that rap was only a segment of. The MCs who contributed the oral aspects of the culture were termed rappers. The hip hop culture was tied to African American neighbourhoods and communities, and the spoken aspect - rap - was a way to effectively communicate cultural ideas. Protest, dreams, rants, hopes - all these emotions have been communicated through rap music.

Hip Hop combined that with the DJs or producers who brought in the musical beats to back tracks. This also led to R&B, blues, and jazz samples being used - tying a modern black culture with an older generations culture. Difference subgenres of hip hop like New Orleans bounce drew from black choir practices such as call and response to modify their particular style of hip hop.

The expression through music meant that communal experiences could be shared via rap music, and disseminated through a wider hip hop culture. It also promoted cultural links between black communities across America, as well as black nationalism. Black nationalism, most commonly associated with the Black Panther Party, focused on the "cultural, political, and economic independence of African Americans", and hip hop culture was a place where all three of these could be achieved. NWA, as an example, was signed to a label established by Eazy E(economic independence, or at least, steps toward it), released their song "Fuck the Police" protesting police brutality, racial profiling, and structural racism(political independence and speech), and promoted hip hop culture while doing so(cultural independence). Historically, rap was only one part of the culture, while hip hop was the overarching term.