The answer to the question of how to distinguish song lyrics from poems is, I think, both straightforward and quite complicated. The straightforward answer is that a poem can exist on the page or in performance without music. It makes its own music, that is to say that the sound, structure and rhythm of the poem are sufficient to produce the poem's overall effect. You could set a good poem to music (and there are many examples of this in the history of music), but then you are making something new. The poem itself doesn't need the music. Lyrics on the other hand, while ideally working well with the music which accompanies them, might struggle to have anywhere near the same effect as poems without music.
So, that's the basic position - but then things get complicated. In reality, we probably have to imagine the relationship between poems and songs lyrics on a sliding scale. At one end, we have lyrics that work wonderfully with the music they were written for, but which would be much less impressive on the page. There are many varieties of this phenomenon, but we could take the example of Paul McCartney's lyrics for 'Yesterday', one of the most successful popular songs ever written, to stand for those lyrics which, on the page, would seem hackneyed and uninspiring, but which are magically transformed in combination with that great tune (and McCartney's brilliance as a performer). Let's face it, rhyming 'yesterday' with 'far away' and 'here to stay' is not the stuff of great poetry. Without the tune (which is pretty hard to forget) it would not be great art. With the tune, it unquestionably is. Another variety of the lyric which can't survive without the tune would be the work of Morrissey. A remarkably lyricist, many of Morrissey's texts actually have relatively few words in them, but these are repeated, woven in and our of the melody and subjected to some of the most extraordinary performances you are likely to encounter. Just think of what he does the with the word 'etcetera' in 'Sweet and Tender Hooligan'. Apart from all of his wit and insight, what makes Morrissey's lyrics work is the way they are sung.
Another interesting sub-category are lyrical collaborations between poets and songwriters. So, for instance, Frank McGuiness has written lyrics for Marianne Faithfull, Paul Muldoon worked with Warren Zevon, and Jeremy Reed collaborated with Marc Almond on Piccadilly Bongo. There are sure to be others out there I don't know. Feel free to leave me a recommendation!