Well--they are kind of sort of the same thing--the piano in "Take Five" is a vamp. Hearing Little Walter behind Muddy singing is a comp. . .
From the Wick:
Comping (an abbreviation of accompanying) is a term used in jazz music to describe the chords, rhythms, and countermelodies that keyboard players (piano or organ) or guitar players use to support a jazz musician's improvised solo or melody lines.
Vamp In music, a vamp is a repeating musical figure, section[8], or accompaniment used in jazz, gospel, soul, and musical theater.[9] Vamps are also found in rock, funk, reggae, R&B, pop, country, and post-sixties jazz.[8] Vamps are usually harmonically spare[8]: A vamp may consist of a single chord or a sequence of chords played in a repeated rhythm. Vamps are generally symmetrical, self-contained, and open to variation.[8] The equivalent in classical music is an ostinato. The equivalent in hip hop is the loop.[8]
The slang term vamp[8] comes from the Middle English word "vampe" (sock), from Old French "avanpie", equivalent to Modern French avant-pied, literally "before-foot".[10]
The term vamp has another meaning in music, which is "to improvise simple accompaniment or variation of a tune." Outside of music, the noun vamp means "something patched up or refurbished" or "something rehashed, as a book based on old material." Similarly, outside of music, the verb "vamp" means "to put together, fabricate or improvise": "With no hard news available about the summit meeting, the reporters vamped up questions based only on rumor."[10] These other meanings are related to the musical meaning, in that a musical vamp is a "fabricated" or "improvised" "rehash" of standard, stock musical phrases.
Classic examples in jazz, include "A Night in Tunisia", "Take Five", "A Love Supreme", "Maiden Voyage", "Cantaloupe Island,"[3] and Chameleon.
My understanding of vamping comes from musical theater. It was like one or two measure phrases that the orchestra pit would repeat over and over until a certain que was delivered so the conductor knew to proceed. It was to ensure that the orchestra didn't get ahead of the cast in terms of where they were in the score.