Typically, assonance is the repetition of a vowel sound to create internal rhyme in a sentence, stanza, etc. whereas, alliteration deals with a group of words that are next to or near each other that start with the same letter (whether a vowel or consonant).
Example of assonance:
“Sarah sat along side Anthony, another one of her awesome friends.”
Here, I repeated the “A’s” and “O’s” and “aw” sounds.
Example of alliteration:
“Trish trash-talked about Tiffany today.”
Here, I repeated the “T’s”