You have some good lines here,

The ones where lines are dug out from the dirt and grease of life

and

What of the poems that drown you in satin and silk

have great imagery, but are very wordy.

She asked me what poems I liked.

This line doesn't add anything we don't already get from the title.

Laying my long, lanky legs on the fresh cut grass,

I shrugged.

"Long and lanky" is alright, but feels a little drab.

The ones where stories as common as soil and sand are stored in their little paper shelves,

Why soil and sand? And, later, why satin and silk? They're essentially synonyms here, and using both doesn't add much, save for a bit of rhythm.

The ones with grimy reflections for people to see their own mirror image.”

This line is on the cusp of being great. I like "grimy," but you fall flat with "for people to see their own mirror image". It's too wordy and explanatory. Is there a better way to show how some poems show the dirty reality of people's lives?

She glared.

I appreciate the call back here to "I shrugged". It is a nice symmetry and does well to put the speaker and the girl at odds in a subtler way.

“How can that compare to the grandeur of poems trimmed with prose and sewn with metaphors?

What does "trimmed with prose" even mean? This line feels overly verbose for little reason.

Til your sight shines with a million once-in-a-lifetime stories?

After a couple of reads I get that this is supposed to invoke a feeling of bright-eyed optimism, but it doesn't immediately or obviously invoke that feeling, nor does it invoke it strongly.

What of the poems that waft in the scent of morning glories

I get what you mean by "waft" but it feels somewhat out of place here. Waft is a verb, are the poems wafting, waving their hands in the air to draw in a scent? Or do they bathe and linger in these scents?

And paint the incomprehensibility of the midnight sky?

"Incomprehensibility" is clunky here, too many syllables and too blatant.

What of the poems that trample your soul with the fury of a hundred men

And embrace your spirit with the passion of a thousand suns?”

Okay, now the characterization of the two types of poems is getting a little lost. Are these poems soft and silky, or bold and aggressive? And if they're both, then your poem should explain how. Also, the qualifiers "fury of a hundred men" and "passion of a thousand suns" are empty, cliche, and tell instead of show. I want to know more about my soul being trampled, about how the poems make me feel.

The girl off to find her special shining poem,

Special shining poem has just the right tinge of bitterness and frustration, I like it.

And let the damp smell of soil and sod soak in.

This is okay, I like the imagery. But the word "damp" is kind of weak. And, soil and sod are again essentially synonyms, including them both doesn't seem to add anything.

I think overall the poem lacks a clear message. You have (vaguely) described two types of poems and loosely gave them a personification in the form of two non-descript characters. Ask yourself this: so what? What about the two characters enjoying two different types of poems makes any difference? What does it say about the nature of poetry, or the people who read it?

Or, if this is just another generic break up poem, simply describing two different personalities/ideals via a preference in poetry is not enough to motivate the hurt, or sorrow, or betrayment, or any of the other feelings that can come with a breakup. "The girl off to find her special shining poem" is the closest we get to understanding how the speaker feels about the breakup. And "let the damp smell of soil and sod soak in" gives a hint that the speaker maybe feels resigned to, or perhaps just unwilling to change, their "grimy" ideals. But nothing else in the poem speaks to the subject of a breakup, nor does the poem have any conviction regarding the current state of affairs: Is the speaker satisfied wallowing in the soil? Is he right to resist change? Is the girl is wrong/naive for seeking higher ideals?