The speakers of both poems are deeply moved by the song of a bird. In Keats's case it is a nightingale, and in Shelley's case it is a skylark.

Keats, however, unlike Shelley, goes into a state of bliss in which he feels he is traveling and almost merging with the nightingale, which he calls an "Immortal" bird. He first says he wishes he could escape the weariness and pain of this world, where people age, or grow frail, and die. Then, he writes,

Away! away! for I will fly to thee,
Not charioted by Bacchus and his pards,
But on the viewless wings of Poesy . . .

In other words, he allows his imagination to take over, and he describes being in the intoxicating world of the nightingale. Near the end of the poem, he must return to the reality of this world, a situation he calls "forlorn." As the poem ends, he wonders if his vision of the bird is a vision or a waking dream, asking, "do I wake or sleep?"

Shelley, in contrast, keeps a distance from the skylark. He observes it rather than merging with it. He finds its song thrilling and moving and wonders what it is thinking as it trills its melodies. Rather than having an experience of ecstatic oneness with the bird, Shelley wants to learn from it how to create the same beautiful poetry as it expresses in its song. Shelly writes,

Teach us, Sprite or Bird,

What sweet thoughts are thine.

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)