Poetry and lyrics are different things because poems aren't written to be sung or to be accompanied by music, and lyrics are. There can be crossover: it often works well to find a poem that's rhythmic, rhymed, or both and put music to it. But that often leads to the addition of refrains and/or other repetition, which aren't nearly as common to poetry. Conversely, if a song lyric is read on paper, especially by someone who doesn't already know the song, it will often read awkwardly, for the simple reason that it wasn't made to be appreciated that way. It was made to be sung and, typically, to be accompanied by at least one instrument. It's often even made (not technically "written" at all in this case) spontaneously while the lyricist plays and sings. Its words therefore inevitably interact with the music, both aiding and depending on it. Just as clearly, the lyrics then owe something to the music, and that something, when subtracted, leaves it an isolated part of an artwork--not the whole thing, which is a song.
Poems have to do the whole thing on their own; sometimes they do well orally, and sometimes they don't at all. "The Love Song of J. Alfred Prufrock," despite its (ironic) name, would make a very difficult song to sing, to accompany, or to follow in sense. E.E. Cummings' poems would lose all of their typographical intricacies if they were made into lyrics. Emily Dickenson's poems would do pretty well, comparatively, because they're typically rhymed and metered, but they're often about four lines long and would invite adaptation via repeated lines or a full repetition of the poem (very common to lyrics; virtually unheard of in poetry). That doesn't mean that some interested musician couldn't do creative things with even Eliot's poems--but the result would then be songs with lyrics and would be judged differently and on different merits. A great poem doesn't necessarily make a great song, and great lyrics definitely don't often stand up as great poems. They don't have to, and the people who made them weren't trying to get them to.