Verse is all on the surface. It has a brassy sheen. Poetry is inward. There's more going on in it than the words declare.
The versifier treats words as counters, the poet treats them as mysteries.
Which isn't to say that poetry is necessarily all mimsy and weird. That was the mistake the late romantics made. The most affecting, the most sublime line in English poetry is, "Pray you, undo this button."
Verse can be intelligent, witty, inventive, memorable- but it never surprises us. The element of surprise is reserved for poetry.
Good verse is better than bad poetry. But, then, "bad poetry" is a contradiction in terms. All poetry is good- and terribly, terribly rare.
Bad poems are like failed attempts at the high jump; we can see the aspiration and applaud the ambition, but if you collide with the bar you collide with the bar.
Bad poetry is good for nothing. Verse on the other hand has a job to do and is to be valued insofar as it does it. Football chants, for example, are mostly very bad verse, but they get the job done. The lousiest football chant, so long as people can be got to sing it, is worth more than a failed poem.
Here are two pieces of verse. They sit side by side on a page of The Oxford Nursery Rhyme Book. One is merely verse. The other is also a poem. I think it's obvious which is which.
Old Boniface he loved good cheer,
And took his glass of Burton,
And when the nights grew sultry hot
He slept without a shirt on.
As I went by a dyer's door
I met a lusty tawnymoor;
Tawny hands, and tawny face,