You've said you're comfortable with the concepts of consonance and assonance, but it might be useful just to recap. Both of these are sound devices, which can be used to create a sense of internal cohesion and make a poem sound more unified when read aloud. They are essentially the same thing but play on different parts of the word.
Consonance, as the name suggests, is when consonants are repeated from one word to another. Usually alliteration (where the first letter of a word is repeated in the next word, or in a word close by ) is considered separate from consonance, but in reality, alliteration can be consonance, too—or assonance. Assonance is when a vowel sound is repeated, and usually it refers to the repetition of the sound in the middle of a word, but technically "ambling around" is both alliteration and assonance, for example.
In this poem, we can find examples of both. "Darling buds," for example, gives us consonance on the letter "d" in a way which brackets the phrase, beginning and ending it with the same letter. We can see assonance in the statement that "fair from fair sometime declines". Note the pleasing long "i" sound being repeated between the final two words, while "fair" is repeated twice—therefore, obviously the same vowel sound is repeated.
We can also identify assonance where different but similar vowel sounds are repeated. For example, in the line beginning "Nor shall death brag," we can find assonance on different types of "a" sound, in the words "shall," "brag," "wanderest" and, arguably, "shade."