Song Lyrics and Poetry are related, but they are very different in many respects. Few serious poems (as opposed to Verse) can be successfully set to music and fewer Song Lyrics can be taken seriously as Poetry when stripped of their Music (Despite examples like Dylan, Cohen, Simon amd Billy Joel, whose lyrics, are considered primarily as "Verse" or "Minor" Poetry, when compared to that of Masters like Yeats, Frost and Eliot) or any Major Poets of Classical Antiquity.
To begin with they have different purposes. Poems are meant to take all forms of experience (both internal and external) and put them in an organized form that will communicate with a reader on a multiple levels, by recreating universal experience so that it might provoke thought or emotion or understanding on multiple levels.
Although Poems can can draw from "Personal" experience, they are more properly suited to dealing with classical questions of Life, Death, History, Conflict, Philosphy, Social Criticism and various moral and ethical issues, and all are most effective if they are presented on a universal level.
The forms that Poems can take are almost unlimited as long as the Basic elements of craftsmanship are employed encompassing Purpose, (whether it be description, analysis , or expression of an emotional or intellectual response to experience, Focus which centers on a single subject and be bounded by Unities of Expression, Thought, and Image which must end in a Denouement or Outcome that Unites the whole.
They can encompass such complexity because they may be examined at leisure and returned to over and over again to capture subtleties of meaning.
Song Lyrics are much more limited as their "primary purpose" is to provide Entertainment and Pleasure by provoking emotional response through describing Personal Interactions, providing Social Criticism, or communicating Feeling.
This may sound roughly similar to the goals of poetry, but "Lyrics", because they must be grasped and absorbed at a rapid pace, are generally limited to a single theme that can be responded to on an emotional or direct level without requiring (or encouraging) thought beyond the basic message, although other levels and any subliminal messages may be generated or otherwise carried by the music itself.
Moreover "Traditional Lyrics' were and are generally encumbered by the necessity of regular Rhythms, Rhyme and a fairly standard over all structure of Verse-Chorus- Bridge-Chorus-Verse-Chorus or variations thereof, which would generally NOT be considered appropriate to Poetry
Some modern lyrics, (particularly HipHop and Rap) are more free form but still subject to basic Rhythms and constrained by the need for multiple internal Rhymes which are often extremely repetitious and even more limited than "Traditional Lyrics" in the subject matter they can effectively express, and even less inclined to multiple levels of meaning.
I write both Songs and Poetry and find that they not only require differences in approach and intent (aside from my own commitment to elements of craftsmanship) but that they seldom can crossover easily from one form to the other, (although I have in fact done so at least a few times in both directions, and found that, except in a couple of cases it took radical revisions in form to do so).
Hope this gives some insight.
JRH