If we were going to describe the sonic qualities of this poem in one word, that word would be repetition. It is everywhere. In rhymes, in words, in sounds—all over the place. You really can't swing a dead cat without hitting some repetition. What? If we know Poe, we think he'd be okay with that metaphor.
The most prominent literary figure in this poem, for example, is anaphora. Take a look at lines 2-3:
As others were—I have not seen
As others saw—I could not bring
The repetition of "As the others [verb]—I could-have not [verb]" really drills these lines into the reader's mind. The same repeated pattern technique can be found in lines 13-15:
From the torrent, or the fountain—
From the red cliff of the mountain—
From the sun that 'round me roll'd.
On top of the anaphora, there is of course the matter of the rhyme scheme. Rhyme is simply the repetition of sounds, and this poem features 11 couplets, or successive lines that rhyme with each other. In addition to the rhymes, the poem repeats lots of other words ("I," "my," "others") and sounds, like the long I sound in lines 17-18:
From the lightning in the sky
As it pass'd me flying by—
The repetition of the same vowel sound here is known in the poetry biz as assonance.
Okay, so there's a lot of sonic repetition going on here. What's the point of it? Well, there are at least two ways one can look at this. First, this poem is all about difference. The speaker feels different from everybody else—he gets sad about different things, likes different things, etc. In a way, the fact that the poem repeats many of the same sounds is kind of like an answer to that theme of difference. All this sameness is the speaker's way of balancing out all that difference.
Still, there's another way you can look at all the sonic repetition. The speaker essentially describes a feeling of being in a rut, right? He's different, he's unique, he's alone, and…well, it's a real bummer. In the same way that the speaker feels sort of stuck, so too the poem is kind of stuck repeating the same words and sounds and structures, over and over and over again.