In terms of rhyme scheme, Keats' ode is ababcdecde. Shelley's ode uses ababb. One differences that stands out between the two is that Shelley uses enjambment, where one thought or sentence continues onto the next line, a kind of continuous stopping and starting, perhaps symbolic of the skylark's wings flapping. Keats' style and content is a bit more personal and therefore his style reflects the shape of his thoughts.
Keats begins "Ode to a Nightingale" in a monologue form, dreaming of drinking his pains away, either through poison or some drug. Then he addresses the nightingale (thou) directly. Shelley addresses the skylark immediately. Both poets express admiration but also envy their muses (the nightingale and the skylark) because they envision the birds' songs as something metaphysical or spiritual.
Keats describes it as "Thou wast not born for death, immortal Bird!" (61). Keats then suggests that this nightingale is the same one described in the Bible and in ancient stories, as if the bird itself was beyond earthly suffering and death, which is what Keats is so desperately trying write or dream his way out of.
Shelley's choice of a skylark is quite useful because it is a bird that only sings while flying, and it usually does so when it is too high to see. So, when Shelley hears the skylark singing, the bird is out of sight. This creates the sensation that the bird really is immaterial, metaphysical, a spiritual voice.
The pale purple even
Melts around thy flight,
Like a star of Heaven
In the broad day-light
Thou art unseen,--but yet I hear thy shrill delight, (16-20).
I think Keats' ode is more full of angst. He was always bothered by thoughts of death and his brother, Tom, had just died of tuberculosis shortly before he wrote this poem. Shelley is plaintive but calm, although he also embraces the spiritual romanticism of the skylark, hoping that it can lead to some higher happiness. He blatantly asks the skylark, "Teach us, Sprite or Bird," (61). You can see this distinction between Keats' paradoxically hopeful angst and Shelley's calm request in the last stanza of each ode.
From Shelley's "To a Skylark," he sounds polite and humble in asking:
Teach me half the gladness
That thy brain must know,
Such harmonious madness
From my lips would flow
The world should listen then-as I am listening now. (101-105)
In the last stanza of Keats' "Ode to a Nightingale," the music has faded. He is dejected, but like Shelley, he believes he might hear the music again and be uplifted if he can just find the right time and/or state of mind in which to have this experience. Keats ends with a much more philosophical question about the experience itself.
Was it a vision, or a waking dream?
Fled is that music:-Do I wake or sleep? (79-80)