Rap (Rapping) is a genre of poetry with rhythmic rhymes that contains thoughts, imagery, metaphors, simile, etc.
Hip-Hop is a culture and sometimes categorized "ghetto" by a lot of people. It's about wearing jewelry, baggy clothes, etc.
Hip-Hop music is something that derived from the Hip-Hop culture.
While Rap still falls under the category of Hip-Hop because it's associated with the culture, Rap is something like a skill.
Nas easily is one of the greatest rapper today with his complex lyrics as opposed to someone like Lil' Wayne or Soulja Boy who cannot actually pull the same lyrics unless you count as "superman that hoe, now watch me youaaauaua" or " shawty wanna thug, bottles in a club". Anyone can pull the same non sensing lyrics like that. Nas however goes something like "But on a positive side, I think Obama provides
Hope - and challenges minds
Of all races and colors to erase the hate
And try and love one another, so many political snakes
We in need of a break, I'm thinkin' I can trust this brotha
But will he keep it way real?
Every innocent n*gga in jail - gets out on appeal
When he wins - will he really care still?
I feel"
Rap is a skill that not any Hip-Hop artist can do. I'll pay $1,000 if Lil' Wayne drop that kind of lyrics without a ghost writer. Now I didn't come up on this my own. A lot of the mainstream hip-hop artist I work with agrees with this (Snoop Dogg, Ice Cube, Rev. Run, Nas, etc.). I am not saying I am not enjoying something from Lil' Wayne, I am just saying there is a difference between those. I hate it when people say rap is crap while it's not even a rap to begin with. Right now I say I can recruit person outside of Los Angeles that can spit random rhyming senseless lyrics and overlay it with a catchy beat, let the studio pay for radio play for 4 months and I guarantee you that that song will hit in the Top 20's in Billboard.