What's the distinction? I listen to a bit (mostly backpack rap stuff like De La and Tribe, plus ultra-mainstream stuff like Jay-Z and Kanye before he started sucking) and before reading this comment I would have assumed the two were synonyms.
Mostly in the eye (or ear) of the beholder. One way to look at it is that rap is something you do, hip-hop is a way you live.
More specifically, hip hop is a cultural phenomenon that developed as an interplay between Jamaican immigrants and the Bronx of the 1970s and 1980s, and includes MCs, DJs, B-Boys and graffiti artists as primary drivers. Rap is a specific element of hip hop, practiced by MCs, in the same way that turntablism (e.g., scratching, mixing, cutting and juggling) is an element practiced by DJs.
Rap is the story telling medium of Hip Hop. Hip Hop is a style of music that can encompass everything from RnB to Soul to even some Electronic Music. Rap is Hip Hop that has a specific emphasis on lyricism, meaning, and narrative. All Rap has elements of Hip Hop, but all Hip Hop does not have elements of Rap. Obviously this is all subjective to a certain degree.
To answer your question, Jay-Z's MCHG is probably more Hip Hop than Rap, while old school Kanye is mos(t) def(initely) Rap. However, unless you plan on getting into conversations like this in the future I really wouldn't worry about the difference.
Ok, this makes sense. I pretty much invariably call the hip-hop/rap music I listen to "hip-hop". De La Soul? Hip-hop. Jay-Z? Hip-hop. The Roots? Hip-hop. Run DMC? Hip-hop.
Even though rap isn't a genre, you can still say "rap music" just like you can say "guitar music". "Hip-hop music", then, is like saying "rock music". Similarly, "Electronic music" (rap) and "Techno" (hip-hop).
rap is the vocal part of it. you can do it to music or not. one can rap on a street corner with no accompaniment whatsoever, or in the shower, etc.
hip hop is the wider genre of music, much of which has rapping in it, but not all of it. there are instrumental hip hop songs, and hip hop songs with only non-rap singing.
kool keith, beastie boys and jurassic 5 come to mind for instrumental tracks, out of the stuff i've heard. that is, hip hop tracks with no rapping.
DJs like z-trip also produce hip-hop backing tracks suitable for rapping over but wouldn't be called rap music by itself.