1727, "resemblance of sounds between words other than rhyme," from French assonance, from assonant, from Latin assonantem (nominative assonans), present participle of assonare/adsonare "to resound, respond," from ad "to" (see ad-) + sonare "to sound" (from PIE root *swen- "to sound").
More specific sense in prosody of "rhyming or correspondence of accented vowels but not consonants" is from 1823. In 20c. the sense tended to merge with consonance in the notion of slant rhyme, off rhyme, but properly there is a distinction.
Assonance is the relationship between words with different consonants immediately preceding and following the last accented vowels, which vowels have identical sounds (hit/will, disturb/bird, absolute/unglued). Consonance is the relationship between words whose final accented vowel sounds are different but with the same consonant frame (truck/trick, billion/bullion, impelling/compiling, trance/trounce). [Miller Williams, "Patterns of Poetry"]