Alliteration is a noun used for the concept of words starting with the same phonetic sound and these words placed in a row or close together.
Alliterate is the verb form of alliteration. To alliterate is to create alliteration either with spoken language or written words. The adjective form is alliterative, and the adverb is alliteratively.
Assonance is the noun used to describe repetitive sounds in words, specifically vowels, which happen at any point in the word. The verb is assonate, though its use is rare and more often it is a misspelling of assassinate. The adjective form is assonant or assonantal.
In short, both alliteration and assonance describe repeating phonetic sounds within words, but alliteration is used for the initial sound while assonance is used for the sound in any other part of the word.
Examples
The various plot twists are dynamic yet daft; kinetic but sometimes kooky; inventive but I’ve run out of alliteration. [National Post]
It helps that the words “latte” and “liberalism” alliterate. [BBC]
Our BGA staff labors over the name issue — Fulton doesn’t feel right — and we eventually settle on Watson, which goes well alliteratively with the word Watchdog. [Chicago Sun Times]
Tigerland’s strength comes not from its lauded “yellow and black” interjection but its clever assonance and internal rhyme and the energy that rolls through the thing like a real song. [The Guardian]
Every bartender has a few drinks in his or her back pocket that can be summoned for that customer who asks for something new, hence the perfectly assonant term “pocket cocktail.” [Washington City Paper]